
     hG                       d dl Z d dlmZ d dlmZmZ d dlZd dlZd dl	m
Z
 d dlmZmZmZ d dlmZ d dlmZ d dlmZ d dlmZ d dlmc mZ d d	lmZmZ d
dlmZ d
dlm Z!m"Z# d
dl$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+ d
dl,m-Z-m.Z.m/Z/ d
dl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6 d dl7m8c m9Z9 d dl:m;Z; d dl<m=Z= d dl>m8Z8 d Z?d Z@d}dZA G d de'          ZB eBddd          ZC G d de'          ZD eDd ddd          ZE G d d e'          ZF eFdd!"          ZG ejH        d#ejI        z            ZJ ejK        eJ          ZLd$ ZMd% ZNd& ZOd' ZPd( ZQd) ZRd* ZSd+ ZT G d, d-e'          ZU eUd./          ZV G d0 d1e'          ZW eWdd2"          ZX G d3 d4e'          ZY eYejI         d5z  ejI        d5z  d6          ZZ G d7 d8e'          Z[ e[ddd9          Z\ G d: d;e]          Z^ G d< d=e=          Z_d> Z`d? Za G d@ dAe'          Zb ebdddB          Zc G dC dDe'          Zd edddE"          Ze G dF dGe'          Zf efdddH          Zg G dI dJe'          Zh ehddK"          Zi G dL dMe'          Zj ejddN"          Zk G dO dPeh          Zl elddQ"          Zm G dR dSe'          Zn endT/          Zo G dU dVe'          Zp epddW"          Zq G dX dYe'          Zr erddZ"          Zs G d[ d\e'          Zt etejI         ejI        d]          Zu G d^ d_e'          Zv evd`/          Zw G da dbe'          Zx exdc/          Zy G dd dee'          Zz ezddf"          Z{ G dg dhe'          Z| e|di/          Z} G dj dke'          Z~ e~ddl"          Z G dm dne'          Z eddo"          Z G dp dqe'          Z eddr"          Z G ds dte'          Z eddu"          Z G dv dwe'          Z eddx"          Z G dy dze'          Z edd{"          Z G d| d}e'          Z edd~"          Z G d de'          Z ed/          Z G d de'          Z edd          Z G d de'          Z ed/          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z ed/          Zd Z G d de'          Z edd"          Z G d de          Z edd"          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z ed/          Z G d de'          Z edd"          Zd Z G d de'          Z ed/          Z G d de'          Z ed/          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z ed/          Z G d de'          Z eddd          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z eddì"          Z G dĄ de'          Z edƬ/          Z G dǄ de'          Z ed dɬ"          Z G dʄ de'          Z eddd̬          Z G d̈́ de'          Z edϬ/          Z G dЄ de'          Z edҬ/          Z G dӄ de'          Z edլ/          Zdք Z G dׄ de'          Z edd٬"          Z G dڄ de'          Z eddܬ          Z G d݄ de'          Z ed߬/          Z G d de'          Z ed/          Z G d de'          Z edd"          Zd Z G d de'          Z edd"          Z G d de'          ZdZ G d deԦ          Z edd"          Z edd"          Zg dZeD ]2Z ej         eeeڦ          deڛ deڛ eզ          Z eeeeݦ           3 G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z ed/          Z G d de'          Z edd"          Z G d  de'          Z ed/          Z G d de'          Z edd"          Zd Z G d de'          Z edd	"          Z G d
 de'          Z edd"          Z G d de'          Z ed/          Z G d de'          Z ed/          Z G d de'          Z edd"          Z G d de'          Z edd"          Z G d de'          Z ed/          Z G d de'          Z eddd          Z G d d e'          Z edd!"          Z G d" d#e'          Z ed$/          Z G d% d&e'          Z  e d'dd(          Z G d) d*e'          Z edd+"          Z G d, d-e'          Z ed./          Z ed//          Z G d0 d1e'          Z edd2"          Z G d3 d4e'          Z	 e	dd5"          Z
 G d6 d7e'          Z ed'dd8          Z G d9 d:e'          Z ed;/          Z G d< d=e'          Z ed>/          Z G d? d@e'          Z edddA          Z edddB          Zej        r
dCe_         G dD dEe'          Z edddF          Z G dG dHe'          Z eddI"          ZdJ ZdK ZdL Z G dM dNe'          Z edOd
P          Z G dQ dRe'          Z eddS"          Z G dT dUe'          Z  e dV/          Z! G dW dXe^          Z" G dY dZe'          Z# e#ddd[          Z$ G d\ d]e'          Z% e%d^/          Z& e%ejI         ejI        d_          Z' G d` dae          Z( e(ddb"          Z) G dc dde'          Z* e*dd#ejI        z  de          Z+ G df dge'          Z, e,dh/          Z- G di dje'          Z. e.d dk"          Z/ G dl dme'          Z0 e0dndop          Z1dq Z2 G dr dse'          Z3 e3dtduddv          Z4 G dw dxe'          Z5 G dy dze'          Z6 e6d{d ej7        |          Z8 e9 e:            ;                                <                                          Z= e%e=e'          \  Z>Z?e>e?z   dxgz   Z@dS (~      N)Iterable)wrapscached_property
Polynomial)extend_notes_in_docstringreplace_notes_in_docstringinherit_docstring_from)LowLevelCallable)optimize)	integrate)_lazyselect
_lazywhere   )_stats)tukeylambda_variancetukeylambda_kurtosis)get_distribution_names	_kurtosisrv_continuous_skew_get_fixed_fit_value_check_shape
_ShapeInfo)kolmognkolmognpkolmogni)_XMIN_EULER_ZETA3_SQRT_PI_SQRT_2_OVER_PI_LOG_SQRT_2_OVER_PI)root_scalar)FitErrorc                     |                      dd           |                      dd           |                      dd           |                      dd           | rt          d| z            dS )a  
    Remove the optimizer-related keyword arguments 'loc', 'scale' and
    'optimizer' from `kwds`.  Then check that `kwds` is empty, and
    raise `TypeError("Unknown arguments: %s." % kwds)` if it is not.

    This function is used in the fit method of distributions that override
    the default method and do not use the default optimization code.

    `kwds` is modified in-place.
    locNscale	optimizermethodUnknown arguments: %s.)pop	TypeError)kwdss    Z/var/www/html/Sam_Eipo/venv/lib/python3.11/site-packages/scipy/stats/_continuous_distns.py_remove_optimizer_parametersr0   &   sz     	HHUDHHWdHH[$HHXt 90478889 9    c                 <     t                      fd            }|S )Nc                     |                     dd                                          }|dk    r( t          t          |           |           j        |i |S  | g|R i |S )Nr*   mle)getlowersupertypefit)selfargsr.   r*   funs       r/   wrapperz _call_super_mom.<locals>.wrapper<   ss    (E**0022U??.5dT**.====3t+d+++d+++r1   )r   )r<   r=   s   ` r/   _call_super_momr>   9   s5     3ZZ, , , , Z, Nr1   c                      |p|dz
  }||z
  } fd} |||          s;|dz  }||z
  }d}t          j        |          rt          |           |||          ;|S )Nr   c                 |    t          j         |                     t          j         |                    k    S Nnpsign)lbrackrbrackr<   s     r/   interval_contains_rootz1_get_left_bracket.<locals>.interval_contains_rootM   s2    wss6{{##rwss6{{';';;;r1      zVThe solver could not find a bracket containing a root to an MLE first order condition.)rC   isinfFitSolverError)r<   rF   rE   diffrG   msgs   `     r/   _get_left_bracketrM   F   s    !vzFF?D< < < < < %$VV44 &	$78F 	& %%% %$VV44 & Mr1   c                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
	ksone_gena  Kolmogorov-Smirnov one-sided test statistic distribution.

    This is the distribution of the one-sided Kolmogorov-Smirnov (KS)
    statistics :math:`D_n^+` and :math:`D_n^-`
    for a finite sample size ``n >= 1`` (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, kstwo, kstest

    Notes
    -----
    :math:`D_n^+` and :math:`D_n^-` are given by

    .. math::

        D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
        D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `ksone` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours
       for probability distribution functions", The Annals of Mathematical
       Statistics, 22(4), pp 592-596 (1951).

    %(example)s

    c                 @    |dk    |t          j        |          k    z  S Nr   rC   roundr:   ns     r/   	_argcheckzksone_gen._argcheck       Q1+,,r1   c                 @    t          dddt          j        fd          gS NrU   Tr   TFr   rC   infr:   s    r/   _shape_infozksone_gen._shape_info       3q"&k=AABBr1   c                 .    t          j        ||           S rA   )scu	_smirnovpr:   xrU   s      r/   _pdfzksone_gen._pdf   s    a####r1   c                 ,    t          j        ||          S rA   )ra   	_smirnovcrc   s      r/   _cdfzksone_gen._cdf   s    }Q"""r1   c                 ,    t          j        ||          S rA   )scsmirnovrc   s      r/   _sfzksone_gen._sf   s    z!Qr1   c                 ,    t          j        ||          S rA   )ra   
_smirnovcir:   qrU   s      r/   _ppfzksone_gen._ppf   s    ~a###r1   c                 ,    t          j        ||          S rA   )rj   smirnoviro   s      r/   _isfzksone_gen._isf       {1a   r1   N)__name__
__module____qualname____doc__rV   r^   re   rh   rl   rq   rt    r1   r/   rO   rO   ]   s        $ $J- - -C C C$ $ $# # #     $ $ $! ! ! ! !r1   rO                 ?ksone)abnamec                   B    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
S )	kstwo_gena  Kolmogorov-Smirnov two-sided test statistic distribution.

    This is the distribution of the two-sided Kolmogorov-Smirnov (KS)
    statistic :math:`D_n` for a finite sample size ``n >= 1``
    (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, ksone, kstest

    Notes
    -----
    :math:`D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
    `kstwo` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided
       Kolmogorov-Smirnov Distribution",  Journal of Statistical Software,
       Vol 39, 11, 1-18 (2011).

    %(example)s

    c                 @    |dk    |t          j        |          k    z  S rQ   rR   rT   s     r/   rV   zkstwo_gen._argcheck   rW   r1   c                 @    t          dddt          j        fd          gS rY   r[   r]   s    r/   r^   zkstwo_gen._shape_info   r_   r1   c                 b    dt          |t                    s|nt          j        |          z  dfS N      ?r|   )
isinstancer   rC   
asanyarrayrT   s     r/   _get_supportzkstwo_gen._get_support   s4    jH55KQQ2=;K;KL 	r1   c                 "    t          ||          S rA   )r   rc   s      r/   re   zkstwo_gen._pdf   s    1~~r1   c                 "    t          ||          S rA   r   rc   s      r/   rh   zkstwo_gen._cdf   s    q!}}r1   c                 &    t          ||d          S NFcdfr   rc   s      r/   rl   zkstwo_gen._sf   s    q!''''r1   c                 &    t          ||d          S )NTr   r   ro   s      r/   rq   zkstwo_gen._ppf   s    1$''''r1   c                 &    t          ||d          S r   r   ro   s      r/   rt   zkstwo_gen._isf   s    1%((((r1   N)rv   rw   rx   ry   rV   r^   r   re   rh   rl   rq   rt   rz   r1   r/   r   r      s        # #H- - -C C C      ( ( (( ( () ) ) ) )r1   r   kstwo)momtyper~   r   r   c                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	kstwobign_gena  Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.

    This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov
    statistic :math:`\sqrt{n} D_n` that measures the maximum absolute
    distance of the theoretical (continuous) CDF from the empirical CDF.
    (see `kstest`).

    %(before_notes)s

    See Also
    --------
    ksone, kstwo, kstest

    Notes
    -----
    :math:`\sqrt{n} D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `kstwobign`  describes the asymptotic distribution (i.e. the limit of
    :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
    empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical
       Distributions",  Ann. Math. Statist. Vol 19, 177-189 (1948).

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zkstwobign_gen._shape_info      	r1   c                 ,    t          j        |           S rA   )ra   _kolmogpr:   rd   s     r/   re   zkstwobign_gen._pdf  s    Qr1   c                 *    t          j        |          S rA   )ra   _kolmogcr   s     r/   rh   zkstwobign_gen._cdf
  s    |Ar1   c                 *    t          j        |          S rA   )rj   
kolmogorovr   s     r/   rl   zkstwobign_gen._sf  s    }Qr1   c                 *    t          j        |          S rA   )ra   	_kolmogcir:   rp   s     r/   rq   zkstwobign_gen._ppf  s    }Qr1   c                 *    t          j        |          S rA   )rj   kolmogir   s     r/   rt   zkstwobign_gen._isf  s    z!}}r1   N)
rv   rw   rx   ry   r^   re   rh   rl   rq   rt   rz   r1   r/   r   r      sy        # #H                       r1   r   	kstwobign)r~   r   rH   c                 H    t          j        | dz   dz            t          z  S NrH          @)rC   exp_norm_pdf_Crd   s    r/   	_norm_pdfr   #  s!    61a4%){**r1   c                 $    | dz   dz  t           z
  S r   )_norm_pdf_logCr   s    r/   _norm_logpdfr   '  s    qD53;''r1   c                 *    t          j        |           S rA   )rj   ndtrr   s    r/   	_norm_cdfr   +  s    71::r1   c                 *    t          j        |           S rA   )rj   log_ndtrr   s    r/   _norm_logcdfr   /  s    ;q>>r1   c                 *    t          j        |           S rA   rj   ndtrirp   s    r/   	_norm_ppfr   3  s    8A;;r1   c                 "    t          |            S rA   r   r   s    r/   _norm_sfr   7  s    aR==r1   c                 "    t          |            S rA   r   r   s    r/   _norm_logsfr   ;  s    r1   c                 "    t          |            S rA   r   r   s    r/   	_norm_isfr   ?  s    aLL=r1   c                       e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Zd Ze eed          d                         Zd ZdS )norm_gena  A normal continuous random variable.

    The location (``loc``) keyword specifies the mean.
    The scale (``scale``) keyword specifies the standard deviation.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norm` is:

    .. math::

        f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   znorm_gen._shape_infoZ  r   r1   Nc                 ,    |                     |          S rA   )standard_normalr:   sizerandom_states      r/   _rvsznorm_gen._rvs]  s    ++D111r1   c                      t          |          S rA   r   r   s     r/   re   znorm_gen._pdf`  s    ||r1   c                      t          |          S rA   )r   r   s     r/   _logpdfznorm_gen._logpdfd      Ar1   c                      t          |          S rA   r   r   s     r/   rh   znorm_gen._cdfg      ||r1   c                      t          |          S rA   r   r   s     r/   _logcdfznorm_gen._logcdfj  r   r1   c                      t          |          S rA   )r   r   s     r/   rl   znorm_gen._sfm  s    {{r1   c                      t          |          S rA   )r   r   s     r/   _logsfznorm_gen._logsfp  s    1~~r1   c                      t          |          S rA   r   r   s     r/   rq   znorm_gen._ppfs  r   r1   c                      t          |          S rA   r   r   s     r/   rt   znorm_gen._isfv  r   r1   c                     dS )N)r{   r|   r{   r{   rz   r]   s    r/   r   znorm_gen._statsy      !!r1   c                 P    dt          j        dt           j        z            dz   z  S Nr   rH   r   rC   logpir]   s    r/   _entropyznorm_gen._entropy|  s     BF1RU7OOA%&&r1   a}          For the normal distribution, method of moments and maximum likelihood
        estimation give identical fits, and explicit formulas for the estimates
        are available.
        This function uses these explicit formulas for the maximum likelihood
        estimation of the normal distribution parameters, so the
        `optimizer` and `method` arguments are ignored.

notesc                    |                     dd           }|                     dd           }t          |           ||t          d          t          j        |          }t          j        |                                          st          d          ||                                }n|}|-t          j        ||z
  dz                                            }n|}||fS )Nflocfscale3All parameters fixed. There is nothing to optimize.$The data contains non-finite values.rH   )	r,   r0   
ValueErrorrC   asarrayisfiniteallmeansqrt)r:   datar.   r   r   r'   r(   s          r/   r9   znorm_gen.fit  s     xx%%(D))$T*** 2  ) * * * z${4  $$&& 	ECDDD<))++CCC>GdSj1_224455EEEEzr1   c                 F    |dz  dk    rt          j        |dz
            S dS )z
        @returns Moments of standard normal distribution for integer n >= 0

        See eq. 16 of https://arxiv.org/abs/1209.4340v2
        rH   r   r   r{   )rj   
factorial2rT   s     r/   _munpznorm_gen._munp  s*     q5A::=Q'''2r1   NN)rv   rw   rx   ry   r^   r   re   r   rh   r   rl   r   rq   rt   r   r   r>   r	   r   r9   r   rz   r1   r/   r   r   C  s,        ,  2 2 2 2                " " "' ' '  6? @ @ @ @ @ _>	 	 	 	 	r1   r   norm)r   c                   D    e Zd ZdZej        Zd Zd Zd Z	d Z
d Zd ZdS )		alpha_gena&  An alpha continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `alpha` ([1]_, [2]_) is:

    .. math::

        f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
                  \exp(-\frac{1}{2} (a-1/x)^2)

    where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.

    `alpha` takes ``a`` as a shape parameter.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
           Distributions, Volume 1", Second Edition, John Wiley and Sons,
           p. 173 (1994).
    .. [2] Anthony A. Salvia, "Reliability applications of the Alpha
           Distribution", IEEE Transactions on Reliability, Vol. R-34,
           No. 3, pp. 251-252 (1985).

    %(example)s

    c                 @    t          dddt          j        fd          gS Nr~   Fr   FFr[   r]   s    r/   r^   zalpha_gen._shape_info      326{NCCDDr1   c                 ^    d|dz  z  t          |          z  t          |d|z  z
            z  S Nr|   rH   )r   r   r:   rd   r~   s      r/   re   zalpha_gen._pdf  s0    AqDz)A,,&y3q5'9'999r1   c                     dt          j        |          z  t          |d|z  z
            z   t          j        t          |                    z
  S )Nr|   )rC   r   r   r   r  s      r/   r   zalpha_gen._logpdf  s>    "&))|l1SU7333bfYq\\6J6JJJr1   c                 L    t          |d|z  z
            t          |          z  S Nr|   r   r  s      r/   rh   zalpha_gen._cdf  s#    3q5!!IaLL00r1   c           
      z    dt          j        |t          j        |t	          |          z            z
            z  S r  )rC   r   rj   r   r   r:   rp   r~   s      r/   rq   zalpha_gen._ppf  s0    2:a9Q<< 8 889999r1   c                 D    t           j        gdz  t           j        gdz  z   S NrH   rC   r\   nanr:   r~   s     r/   r   zalpha_gen._stats  s    xzRVHQJ&&r1   N)rv   rw   rx   ry   r   _open_support_mask_support_maskr^   re   r   rh   rq   r   rz   r1   r/   r   r     s         > "4ME E E: : :K K K1 1 1: : :' ' ' ' 'r1   r   alphac                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	
anglit_gena  An anglit continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `anglit` is:

    .. math::

        f(x) = \sin(2x + \pi/2) = \cos(2x)

    for :math:`-\pi/4 \le x \le \pi/4`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zanglit_gen._shape_info  r   r1   c                 0    t          j        d|z            S r	  )rC   cosr   s     r/   re   zanglit_gen._pdf  s    vac{{r1   c                 P    t          j        |t           j        dz  z             dz  S )N   r   rC   sinr   r   s     r/   rh   zanglit_gen._cdf	  s!    vaai  #%%r1   c                 n    t          j        t          j        |                    t           j        dz  z
  S Nr  )rC   arcsinr   r   r   s     r/   rq   zanglit_gen._ppf  s%    y$$RU1W,,r1   c                     dt           j        t           j        z  dz  dz
  ddt           j        dz  dz
  z  t           j        t           j        z  dz
  dz  z  fS )	Nr{      r   r  r  `      rH   rC   r   r]   s    r/   r   zanglit_gen._stats  sH    BE"%KN3&RB-?ruQQR@R-RRRr1   c                 0    dt          j        d          z
  S Nr   rH   rC   r   r]   s    r/   r   zanglit_gen._entropy      {r1   N
rv   rw   rx   ry   r^   re   rh   rq   r   r   rz   r1   r/   r  r    s{         &    & & &- - -S S S    r1   r  r  anglitc                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	arcsine_gena  An arcsine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `arcsine` is:

    .. math::

        f(x) = \frac{1}{\pi \sqrt{x (1-x)}}

    for :math:`0 < x < 1`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zarcsine_gen._shape_info-  r   r1   c                     t          j        d          5  dt           j        z  t          j        |d|z
  z            z  cd d d            S # 1 swxY w Y   d S )Nignoredivider|   r   )rC   errstater   r   r   s     r/   re   zarcsine_gen._pdf0  s    [))) 	. 	.ru9RWQ!W---	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	. 	.s   *AAAc                 n    dt           j        z  t          j        t          j        |                    z  S Nr   )rC   r   r  r   r   s     r/   rh   zarcsine_gen._cdf5  s%    25y271::....r1   c                 P    t          j        t           j        dz  |z            dz  S r0  r  r   s     r/   rq   zarcsine_gen._ppf8  s!    vbeCik""C''r1   c                     d}d}d}d}||||fS )Nr   g      ?r         rz   r:   mumu2g1g2s        r/   r   zarcsine_gen._stats;  s$    3Br1   c                     dS )Ngοrz   r]   s    r/   r   zarcsine_gen._entropyB  s    &&r1   Nr%  rz   r1   r/   r(  r(    sx         &  . . .
/ / /( ( (  ' ' ' ' 'r1   r(  arcsinec                       e Zd ZdZd ZdS )FitDataErrorz=Raised when input data is inconsistent with fixed parameters.c                 B    d                     |||          f| _        d S )NzInvalid values in `data`.  Maximum likelihood estimation with {distr!r} requires that {lower!r} < (x - loc)/scale  < {upper!r} for each x in `data`.)distrr6   upper)formatr;   )r:   r>  r6   r?  s       r/   __init__zFitDataError.__init__N  s4    AAG5 BH B7 B7
			r1   Nrv   rw   rx   ry   rA  rz   r1   r/   r<  r<  I  s)        GG
 
 
 
 
r1   r<  c                       e Zd ZdZd ZdS )rJ   zN
    Raised when a solver fails to converge while fitting a distribution.
    c                 L    d}||                     dd          z  }|f| _        d S )Nz1Solver for the MLE equations failed to converge: 
 )replacer;   )r:   mesgemsgs      r/   rA  zFitSolverError.__init__]  s,    BT2&&&G			r1   NrB  rz   r1   r/   rJ   rJ   W  s-         
    r1   rJ   c                 p    t          j        | |z             }||| t          j        |           z   z  z
  }|S rA   rj   psi)r~   r   rU   s1psiabfuncs         r/   _beta_mle_arP  c  s8     F1q5MMEeVbfQii'((DKr1   c                     | \  }}t          j        ||z             }||| t          j        |          z   z  z
  ||| t          j        |          z   z  z
  g}|S rA   rK  )thetarU   rM  s2r~   r   rN  rO  s           r/   _beta_mle_abrT  l  sb     DAqF1q5MMEufrvayy())ufrvayy())+DKr1   c                        e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Z fdZe eed           fd                        Z xZS )beta_gena  A beta continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `beta` is:

    .. math::

        f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
                          {\Gamma(a) \Gamma(b)}

    for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `beta` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS Nr~   Fr   r   r   r[   r:   iaibs      r/   r^   zbeta_gen._shape_info  =    UQK@@UQK@@Bxr1   Nc                 0    |                     |||          S rA   beta)r:   r~   r   r   r   s        r/   r   zbeta_gen._rvs  s      At,,,r1   c                 .    t          j        |||          S rA   )_boost	_beta_pdfr:   rd   r~   r   s       r/   re   zbeta_gen._pdf  s     1a(((r1   c                     t          j        |dz
  |           t          j        |dz
  |          z   }|t          j        ||          z  }|S r  )rj   xlog1pyxlogybetaln)r:   rd   r~   r   lPxs        r/   r   zbeta_gen._logpdf  sG    jS1"%%S!(<(<<ryA
r1   c                 .    t          j        |||          S rA   )ra  	_beta_cdfrc  s       r/   rh   zbeta_gen._cdf  s    1a(((r1   c                 .    t          j        |||          S rA   )ra  _beta_sfrc  s       r/   rl   zbeta_gen._sf  s    q!Q'''r1   c                     t          j                    5  d}t          j        d|           t          j        |||          cd d d            S # 1 swxY w Y   d S )Nz!overflow encountered in _beta_isfr+  message)warningscatch_warningsfilterwarningsra  	_beta_isf)r:   rd   r~   r   ro  s        r/   rt   zbeta_gen._isf  s    $&& 	- 	-9G#Hg>>>>#Aq!,,		- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	-   .AAAc                     t          j                    5  d}t          j        d|           t          j        |||          cd d d            S # 1 swxY w Y   d S )Nz!overflow encountered in _beta_ppfr+  rn  )rp  rq  rr  ra  	_beta_ppf)r:   rp   r~   r   ro  s        r/   rq   zbeta_gen._ppf  s    $&& 	- 	-9G#Hg>>>>#Aq!,,	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	- 	-rt  c                     t          j        ||          t          j        ||          t          j        ||          t          j        ||          fS rA   )ra  
_beta_mean_beta_variance_beta_skewness_beta_kurtosis_excessr:   r~   r   s      r/   r   zbeta_gen._stats  sL    a##!!Q''!!Q''(A..	0 	0r1   c                     t          |          t          |          fd}t          j        |d          \  }}t	                                          |||f          S )Nc                 F   | \  }}d||z
  z  t          j        ||z   dz             z  ||z   dz   z  t          j        ||z            z  }|dz  |dz  d|z  dz
  z  z
  |dz  |dz   z  z   d|z  |z  |dz   z  z
  }|||z  ||z   dz   z  ||z   dz   z  z  }|dz  }|z
  |z
  gS )NrH   r         rC   r   )rd   r~   r   skkur7  r8  s        r/   rO  z beta_gen._fitstart.<locals>.func  s    DAqAaCQ+++q1uqy9BGAaCLLHBA1ac!e$q!tQqSz1AaCE1Q3K?B!A#qs1u+qs1u%%B!GBrE2b5>!r1   )r|   r|   r;   )r   r   r   fsolver7   	_fitstart)r:   r   rO  r~   r   r7  r8  	__class__s        @@r/   r  zbeta_gen._fitstart  sq    4[[t__	" 	" 	" 	" 	" 	" tZ001ww  QF 333r1   z        In the special case where `method="MLE"` and
        both `floc` and `fscale` are given, a
        `ValueError` is raised if any value `x` in `data` does not satisfy
        `floc < x < floc + fscale`.

r   c           	         |                     dd           }|                     dd           }|| t                      j        |g|R i |S |                    dd            |                    dd            t	          |g d          }t	          |g d          }t          |           ||t          d          t          j        |          	                                st          d          t          j
        |          |z
  |z  }t          j        |dk              st          j        |dk              rt          d	|||z   
          |                                }||||}	d|z
  }d|z
  }n|}	|	|z  d|z
  z  }
t          j        t           |
|	t#          |          t          j        |                                          fd          \  }}}}|dk    rt)          |          |d         }
||	|
}	}
nt          j        |                                          }t+          j        |                                           }|d|z
  z  |                    d          z  dz
  }||z  }
d|z
  |z  }	t          j        t0          |
|	gt#          |          ||fd          \  }}}}|dk    rt)          |          |\  }
}	|
|	||fS )Nr   r   f0fafix_a)f1fbfix_br   r   r   r   r_  r6   r?  T)r;   full_output)rH  )ddof)r5   r7   r9   r,   r   r0   r   rC   r   r   ravelanyr<  r   r   r  rP  lenr   sumrJ   rj   log1pvarrT  )r:   r   r;   r.   r   r   r  r  xbarr   r~   rR  infoierrH  rM  rS  facr  s                     r/   r9   zbeta_gen.fit  s'    xx%%(D))<6>577;t3d333d333 	4   !$(=(=(=>>!$(=(=(=>>$T***>bn ) * * * {4  $$&& 	ECDDD %/6$!) 	Htqy 1 1 	HvTGGGGyy{{>R^ ~ 4x4x DAH%A &._QTBF4LL$4$4$6$67 & & &"E4d
 axx$$////aA~ !1 !!##B4%$$&&B !d(#dhhAh&6&66:Cs
ATS A &._q!f$iiR( & & &"E4d
 axx$$////DAq!T6!!r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   rt   rq   r   r  r>   r   r   r9   __classcell__r  s   @r/   rV  rV  z  s"        .  
- - - -) ) )  
) ) )( ( (- - -- - -0 0 04 4 4 4 4 } 5+ , , ,
c" c" c" c", , _c" c" c" c" c"r1   rV  r_  c                   F    e Zd ZdZej        Zd Zd	dZd Z	d Z
d Zd ZdS )
betaprime_gena  A beta prime continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `betaprime` is:

    .. math::

        f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}

    for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).

    `betaprime` takes ``a`` and ``b`` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS rX  r[   rY  s      r/   r^   zbetaprime_gen._shape_infoU  r\  r1   Nc                     t                               |||          }t                               |||          }||z  S Nr   r   )gammarvs)r:   r~   r   r   r   u1u2s          r/   r   zbetaprime_gen._rvsZ  s9    YYqt,Y??YYqt,Y??Bwr1   c                 T    t          j        |                     |||                    S rA   rC   r   r   rc  s       r/   re   zbetaprime_gen._pdf_  "    vdll1a++,,,r1   c                     t          j        |dz
  |          t          j        ||z   |          z
  t          j        ||          z
  S r  )rj   rf  re  rg  rc  s       r/   r   zbetaprime_gen._logpdfc  s<    xC##bjQ&:&::RYq!__LLr1   c                 :    t          j        |||d|z   z            S r  )rj   betaincrc  s       r/   rh   zbetaprime_gen._cdff  s    z!Q2a4)))r1   c                     |dk    r*t          j        |dk    ||dz
  z  t           j                  S |dk    r6t          j        |dk    ||dz   z  |dz
  |dz
  z  z  t           j                  S |dk    rBt          j        |dk    ||dz   z  |dz   z  |dz
  |dz
  z  |dz
  z  z  t           j                  S |dk    rNt          j        |dk    ||dz   z  |dz   z  |dz   z  |dz
  |dz
  z  |dz
  z  |dz
  z  z  t           j                  S t          )	Nr|   r   r   rH         @r        @r  )rC   wherer\   NotImplementedErrorr:   rU   r~   r   s       r/   r   zbetaprime_gen._munpi  sJ   888AEquIF$ $ $ #XX8AEquI##7F$ $ $ #XX8AEquIqu-###/FGF$ $ $ #XX8AEC[!c'2AG< 3wS11s7;QWEGF$ $ $
 &%r1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   rh   r   rz   r1   r/   r  r  <  s         , "4M  
   
- - -M M M* * *& & & & &r1   r  	betaprimec                   8    e Zd ZdZd Zd Zd Zd Zd
dZd Z	d	S )bradford_genab  A Bradford continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `bradford` is:

    .. math::

        f(x, c) = \frac{c}{\log(1+c) (1+cx)}

    for :math:`0 <= x <= 1` and :math:`c > 0`.

    `bradford` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS NcFr   r   r[   r]   s    r/   r^   zbradford_gen._shape_info  r   r1   c                 B    |||z  dz   z  t          j        |          z  S r  rj   r  r:   rd   r  s      r/   re   zbradford_gen._pdf  s!    AaC#I!,,r1   c                 Z    t          j        ||z            t          j        |          z  S rA   r  r  s      r/   rh   zbradford_gen._cdf  s!    x!}}rx{{**r1   c                 Z    t          j        |t          j        |          z            |z  S rA   rj   expm1r  r:   rp   r  s      r/   rq   zbradford_gen._ppf  s#    xBHQKK((1,,r1   mvc                 d   t          j        d|z             }||z
  ||z  z  }|dz   |z  d|z  z
  d|z  |z  |z  z  }d }d }d|v ryt          j        d          d|z  |z  d|z  |z  |dz   z  z
  d|z  |z  ||dz   z  dz   z  z   z  }|t          j        |||dz
  z  d|z  z   z            d|z  |dz
  z  d|z  z   z  z  }d	|v rj|dz  |dz
  z  |d|z  d
z
  z  dz   z  d|z  |z  |z  |dz
  z  |dz
  z  z   d|z  |z  |z  d|z  dz
  z  z   d|dz  z  z   }|d|z  ||dz
  z  d|z  z   dz  z  z  }||||fS )Nr|   r   rH   s   	   r  r  kr     r     )rC   r   r   )r:   r  momentsr  r5  r6  r7  r8  s           r/   r   zbradford_gen._stats  s   F3q5MMcAaC[#qyQ1Qq)'>>RT!VAaCE1Q3K/!Aq!A#wqy0AABB"'!Q!WQqS[/**AaC1IacM::B'>>Q$!*a1Rjm,RT!VAXqs^QqS-AAA#a%'1Q3r6"#%'1W-B!A#q!A#wqs{Q&&&B3Br1   c                 j    t          j        d|z             }|dz  t          j        ||z            z
  S Nr   r   r#  )r:   r  r  s      r/   r   zbradford_gen._entropy  s.    F1Q3KKurvac{{""r1   Nr  r%  rz   r1   r/   r  r    s         *E E E- - -+ + +- - -   # # # # #r1   r  bradfordc                   N    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd ZdS )burr_gena  A Burr (Type III) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr12 : Burr Type XII distribution
    mielke : Mielke Beta-Kappa / Dagum distribution

    Notes
    -----
    The probability density function for `burr` is:

    .. math::

        f(x; c, d) = c d \frac{x^{-c - 1}}
                              {{(1 + x^{-c})}^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and
    :math:`d`.

    This is the PDF corresponding to the third CDF given in Burr's list;
    specifically, it is equation (11) in Burr's paper [1]_. The distribution
    is also commonly referred to as the Dagum distribution [2]_. If the
    parameter :math:`c < 1` then the mean of the distribution does not
    exist and if :math:`c < 2` the variance does not exist [2]_.
    The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.

    %(after_notes)s

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).
    .. [2] https://en.wikipedia.org/wiki/Dagum_distribution
    .. [3] Kleiber, Christian. "A guide to the Dagum distributions."
       Modeling Income Distributions and Lorenz Curves  pp 97-117 (2008).

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS Nr  Fr   r   dr[   r:   icids      r/   r^   zburr_gen._shape_info  r\  r1   c                 d    t          |dk    |||gd d           }|j        dk    r|d         S |S )Nr   c                 6    ||z  | ||z  dz
  z  z  d| |z  z   z  S rQ   rz   x_c_d_s      r/   <lambda>zburr_gen._pdf.<locals>.<lambda>  s(    b2gbeAg&?1r2v:&N r1   c                 @    ||z  | | dz
  z  z  d| | z  z   |dz   z  z  S Nr|   r   rz   r  s      r/   r  zburr_gen._pdf.<locals>.<lambda>  s6    BGrrcCi7H,I./""+o28-L-N r1   f2rz   r   ndimr:   rd   r  r  outputs        r/   re   zburr_gen._pdf  sW    AFQ1INNO OP P P ;!":r1   c                 d    t          |dk    |||gd d           }|j        dk    r|d         S |S )Nr   c                     t          j        |          t          j        |          z   t          j        ||z  dz
  |           z   |dz   t          j        | |z            z  z
  S rQ   )rC   r   rj   rf  r  r  s      r/   r  z"burr_gen._logpdf.<locals>.<lambda>  sS    r

RVBZZ 7"(2b519b:Q:Q Q#%a428BH+=+="=!> r1   c                     t          j        |          t          j        |          z   t          j        | dz
  |           z   t          j        |dz   | | z            z
  S rQ   rC   r   rj   rf  re  r  s      r/   r  z"burr_gen._logpdf.<locals>.<lambda>  sT    RVBZZ"&**%<')xa'<'<&=')z"Q$bS	'B'B&C r1   r  rz   r  r  s        r/   r   zburr_gen._logpdf  s\    FQ1I? ?D D	E E E ;!":r1   c                     d|| z  z   | z  S rQ   rz   r:   rd   r  r  s       r/   rh   zburr_gen._cdf  s    AGr""r1   c                 :    t          j        || z            | z  S rA   r  r  s       r/   r   zburr_gen._logcdf  s    xQB  QB''r1   c                 T    t          j        |                     |||                    S rA   rC   r   r   r  s       r/   rl   zburr_gen._sf  "    vdkk!Q**+++r1   c                 B    t          j        d|| z  z   | z             S rQ   rC   r  r  s       r/   r   zburr_gen._logsf  s&    x1qA2w;1"--...r1   c                 $    |d|z  z  dz
  d|z  z  S N      r   rz   r:   rp   r  r  s       r/   rq   zburr_gen._ppf  s    DFa46**r1   c           	         t          j        dd                              dd          |z  }t          j        ||z   d|z
            |z  \  }}}}t          j        |dk    |t           j                  }||dz  z
  }	t          j        |dk    |	t           j                  }
t          |dk    |||||	fd t           j        	          }t          |d
k    ||||||	fd t           j        	          }t          j        |          dk    rN|	                                |
	                                |	                                |	                                fS ||
||fS )Nr      r  r|   rH   r   r  c                 Z    |d|z  |z  z
  d|dz  z  z   t          j        |dz            z  S )Nr  rH   r  )r  e1e2e3mu2_if_cs        r/   r  z!burr_gen._stats.<locals>.<lambda>!  s4    R!B$r'\Ab!eG-CrwPX[\}G]G],] r1   	fillvaluer  c                 T    |d|z  |z  z
  d|z  |dz  z  z   d|dz  z  z
  |dz  z  dz
  S )Nr  r  rH   r  rz   )r  r  r  r  e4r  s         r/   r  z!burr_gen._stats.<locals>.<lambda>&  sB    qtBw,2b!e+aAg51DI r1   r   )
rC   arangereshaperj   r_  r  r  r   r  item)r:   r  r  ncr  r  r  r  r5  r  r6  r7  r8  s                r/   r   zburr_gen._stats  sM   Yq!__$$Qq))A-Rb11A5BBXa#gr26**A:hq3w"&11GBH%]]f	  
 GBB)K Kf   71::??7799chhjj"''))RWWYY>>3Br1   c                     d t          j        |          t          j        |          t          j        |          }}}t          ||k    ||k    z  ||k    z  |||ffdt           j                  S )Nc                 N    d| z  |z  }|t          j        d|z
  ||z             z  S r  rj   r_  )rU   r  r  r  s       r/   __munpzburr_gen._munp.<locals>.__munp.  s.    a!BrwsRxR0000r1   c                      || |          S rA   rz   )r  r  rU   _burr_gen__munps      r/   r  z burr_gen._munp.<locals>.<lambda>3  s    &&Aq// r1   )rC   r   r   r  )r:   rU   r  r  r  s       @r/   r   zburr_gen._munp-  s|    	1 	1 	1 *Q--A
1a11q5Q!V,Q7!Q9999&" " 	"r1   N)rv   rw   rx   ry   r^   re   r   rh   r   rl   r   rq   r   r   rz   r1   r/   r  r    s        + +`  
  
 
 
# # #( ( (, , ,/ / /+ + +  ," " " " "r1   r  burrc                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZdS )
burr12_gena}  A Burr (Type XII) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr : Burr Type III distribution

    Notes
    -----
    The probability density function for `burr12` is:

    .. math::

        f(x; c, d) = c d \frac{x^{c-1}}
                              {(1 + x^c)^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
    and :math:`d`.

    This is the PDF corresponding to the twelfth CDF given in Burr's list;
    specifically, it is equation (20) in Burr's paper [1]_.

    %(after_notes)s

    The Burr type 12 distribution is also sometimes referred to as
    the Singh-Maddala distribution from NIST [2]_.

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).

    .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm

    .. [3] "Burr distribution",
       https://en.wikipedia.org/wiki/Burr_distribution

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS r  r[   r  s      r/   r^   zburr12_gen._shape_infog  r\  r1   c                 T    t          j        |                     |||                    S rA   r  r  s       r/   re   zburr12_gen._pdfl  r  r1   c                     t          j        |          t          j        |          z   t          j        |dz
  |          z   t          j        | dz
  ||z            z   S rQ   r  r  s       r/   r   zburr12_gen._logpdfp  sN    vayy26!99$rxAq'9'99BJr!tQPQT<R<RRRr1   c                 V    t          j        |                     |||                     S rA   )rj   r  r   r  s       r/   rh   zburr12_gen._cdfs  s%    Q1--....r1   c                 @    t          j        d||z  z   | z             S rQ   r  r  s       r/   r   zburr12_gen._logcdfv  s$    x!ad(qb))***r1   c                 T    t          j        |                     |||                    S rA   r  r  s       r/   rl   zburr12_gen._sfy  r  r1   c                 4    t          j        | ||z            S rA   rj   re  r  s       r/   r   zburr12_gen._logsf|  s    z1"ad###r1   c                 h    t          j        d|z  t          j        |           z            d|z  z  S )Nr   r  r  s       r/   rq   zburr12_gen._ppf  s0     x1rx||+,,qs33r1   c                 N    d|z  |z  }|t          j        d|z   ||z
            z  S r  r  )r:   rU   r  r  r  s        r/   r   zburr12_gen._munp  s.    !VaZ2738QV,,,,r1   N)rv   rw   rx   ry   r^   re   r   rh   r   rl   r   rq   r   rz   r1   r/   r  r  :  s        + +X  
- - -S S S/ / /+ + +, , ,$ $ $4 4 4- - - - -r1   r  burr12c                   T    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd ZdS )fisk_gena  A Fisk continuous random variable.

    The Fisk distribution is also known as the log-logistic distribution.

    %(before_notes)s

    See Also
    --------
    burr

    Notes
    -----
    The probability density function for `fisk` is:

    .. math::

        f(x, c) = \frac{c x^{c-1}}
                       {(1 + x^c)^2}

    for :math:`x >= 0` and :math:`c > 0`.

    Please note that the above expression can be transformed into the following
    one, which is also commonly used:

    .. math::

        f(x, c) = \frac{c x^{-c-1}}
                       {(1 + x^{-c})^2}

    `fisk` takes ``c`` as a shape parameter for :math:`c`.

    `fisk` is a special case of `burr` or `burr12` with ``d=1``.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zfisk_gen._shape_info  r   r1   c                 :    t                               ||d          S r  )r  re   r  s      r/   re   zfisk_gen._pdf  s    yyAs###r1   c                 :    t                               ||d          S r  )r  rh   r  s      r/   rh   zfisk_gen._cdf      yyAs###r1   c                 :    t                               ||d          S r  )r  rl   r  s      r/   rl   zfisk_gen._sf  s    xx1c"""r1   c                 :    t                               ||d          S r  )r  r   r  s      r/   r   zfisk_gen._logpdf  s    ||Aq#&&&r1   c                 :    t                               ||d          S r  )r  r   r  s      r/   r   zfisk_gen._logcdf  s    ||Aq#&&&r1   c                 :    t                               ||d          S r  )r  r   r  s      r/   r   zfisk_gen._logsf  s    {{1a%%%r1   c                 :    t                               ||d          S r  )r  rq   r  s      r/   rq   zfisk_gen._ppf  r   r1   c                 :    t                               ||d          S r  )r  r   r:   rU   r  s      r/   r   zfisk_gen._munp  s    zz!Q$$$r1   c                 8    t                               |d          S r  )r  r   r:   r  s     r/   r   zfisk_gen._stats  s    {{1c"""r1   c                 0    dt          j        |          z
  S r	  r#  r)  s     r/   r   zfisk_gen._entropy      26!99}r1   N)rv   rw   rx   ry   r^   re   rh   rl   r   r   r   rq   r   r   r   rz   r1   r/   r  r    s        % %LE E E$ $ $$ $ $# # #' ' '' ' '& & &$ $ $% % %# # #    r1   r  fiskc                   J    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 ZddZd
S )
cauchy_gena	  A Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `cauchy` is

    .. math::

        f(x) = \frac{1}{\pi (1 + x^2)}

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zcauchy_gen._shape_info  r   r1   c                 2    dt           j        z  d||z  z   z  S r  r   r   s     r/   re   zcauchy_gen._pdf      25y#ac'""r1   c                 P    ddt           j        z  t          j        |          z  z   S r   rC   r   arctanr   s     r/   rh   zcauchy_gen._cdf       SYry||+++r1   c                 d    t          j        t           j        |z  t           j        dz  z
            S r0  rC   tanr   r   s     r/   rq   zcauchy_gen._ppf  s#    vbeAgbeCi'(((r1   c                 P    ddt           j        z  t          j        |          z  z
  S r   r3  r   s     r/   rl   zcauchy_gen._sf  r5  r1   c                 d    t          j        t           j        dz  t           j        |z  z
            S r0  r7  r   s     r/   rt   zcauchy_gen._isf  s#    vbeCia'(((r1   c                 ^    t           j        t           j        t           j        t           j        fS rA   rC   r  r]   s    r/   r   zcauchy_gen._stats      vrvrvrv--r1   c                 D    t          j        dt           j        z            S r  r   r]   s    r/   r   zcauchy_gen._entropy      vagr1   Nc                 L    t          j        |g d          \  }}}|||z
  dz  fS )N   2   K   rH   rC   
percentile)r:   r   r;   p25p50p75s         r/   r  zcauchy_gen._fitstart  s0    dLLL99S#S3YM!!r1   rA   )rv   rw   rx   ry   r^   re   rh   rq   rl   rt   r   r   r  rz   r1   r/   r.  r.    s         &  # # #, , ,) ) ), , ,) ) ). . .  " " " " " "r1   r.  cauchyc                   J    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd ZdS )chi_gena  A chi continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `chi` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
                   x^{k-1} \exp \left( -x^2/2 \right)

    for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation). :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    Special cases of `chi` are:

        - ``chi(1, loc, scale)`` is equivalent to `halfnorm`
        - ``chi(2, 0, scale)`` is equivalent to `rayleigh`
        - ``chi(3, 0, scale)`` is equivalent to `maxwell`

    `chi` takes ``df`` as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS NdfFr   r   r[   r]   s    r/   r^   zchi_gen._shape_info0      4BF^DDEEr1   Nc                 `    t          j        t                              |||                    S r  )rC   r   chi2r  r:   rO  r   r   s       r/   r   zchi_gen._rvs3  s$    wtxxLxIIJJJr1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   rO  s      r/   re   zchi_gen._pdf6  s"     vdll1b))***r1   c                     t          j        d          dt          j        d          z  |z  z
  t          j        d|z            z
  }|t          j        |dz
  |          z   d|dz  z  z
  S )NrH   r   r|   )rC   r   rj   gammalnrf  )r:   rd   rO  ls       r/   r   zchi_gen._logpdf<  s_    F1II26!99R'"*RU*;*;;28BGQ'''"QT'11r1   c                 >    t          j        d|z  d|dz  z            S Nr   rH   rj   gammaincrU  s      r/   rh   zchi_gen._cdf@  s     {2b5"QT'***r1   c                 >    t          j        d|z  d|dz  z            S rZ  rj   	gammainccrU  s      r/   rl   zchi_gen._sfC  s     |BrE2ad7+++r1   c                 \    t          j        dt          j        d|z  |          z            S NrH   r   rC   r   rj   gammaincinvr:   rp   rO  s      r/   rq   zchi_gen._ppfF  s'    wq2q111222r1   c                 \    t          j        dt          j        d|z  |          z            S ra  rC   r   rj   gammainccinvrd  s      r/   rt   zchi_gen._isfI  s'    wqB222333r1   c                    t          j        d          t          j        t          j        |dz  dz             t          j        |dz            z
            z  }|||z  z
  }d|dz  z  |dd|z  z
  z  z   t          j        t          j        |d                    z  }d|z  d|z
  z  d|d	z  z  z
  d	|dz  z  d|z  dz
  z  z   }|t          j        |dz            z  }||||fS )
NrH   r   r   r  r         ?r|   r  r  )rC   r   r   rj   rW  r   powerr:   rO  r5  r6  r7  r8  s         r/   r   zchi_gen._statsL  s    WQZZrz"S&*55bjC6H6HHIII2b5jCi"a"f+%rz"(32D2D'E'EErT3r6]1RU7"Qr1uW"Q%77
bjc"""3Br1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   rq   rt   r   rz   r1   r/   rL  rL    s         <F F FK K K K+ + +2 2 2+ + +, , ,3 3 34 4 4    r1   rL  chic                   J    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd ZdS )chi2_gena  A chi-squared continuous random variable.

    For the noncentral chi-square distribution, see `ncx2`.

    %(before_notes)s

    See Also
    --------
    ncx2

    Notes
    -----
    The probability density function for `chi2` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
                   x^{k/2-1} \exp \left( -x/2 \right)

    for :math:`x > 0`  and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation).

    `chi2` takes ``df`` as a shape parameter.

    The chi-squared distribution is a special case of the gamma
    distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and
    ``scale = 2``.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS rN  r[   r]   s    r/   r^   zchi2_gen._shape_infoz  rP  r1   Nc                 .    |                     ||          S rA   )	chisquarerS  s       r/   r   zchi2_gen._rvs}  s    %%b$///r1   c                 R    t          j        |                     ||                    S rA   r  rU  s      r/   re   zchi2_gen._pdf  s     vdll1b))***r1   c                     t          j        |dz  dz
  |          |dz  z
  t          j        |dz            z
  t          j        d          |z  dz  z
  S )Nr   r   rH   )rj   rf  rW  rC   r   rU  s      r/   r   zchi2_gen._logpdf  sO    x2a##ad*RZ2->->>"&))B,PRARRRr1   c                 ,    t          j        ||          S rA   )rj   chdtrrU  s      r/   rh   zchi2_gen._cdf      xAr1   c                 ,    t          j        ||          S rA   )rj   chdtrcrU  s      r/   rl   zchi2_gen._sf      yQr1   c                 ,    t          j        ||          S rA   )rj   chdtrir:   prO  s      r/   rt   zchi2_gen._isf  ry  r1   c                 8    dt          j        |dz  |          z  S r	  rj   rc  r|  s      r/   rq   zchi2_gen._ppf  s    1a((((r1   c                 Z    |}d|z  }dt          j        d|z            z  }d|z  }||||fS )NrH   r         (@r  rk  s         r/   r   zchi2_gen._stats  s=    drws2v"W3Br1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   rt   rq   r   rz   r1   r/   rn  rn  X  s           BF F F0 0 0 0+ + +S S S            ) ) )    r1   rn  rR  c                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZdS )
cosine_gena\  A cosine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The cosine distribution is an approximation to the normal distribution.
    The probability density function for `cosine` is:

    .. math::

        f(x) = \frac{1}{2\pi} (1+\cos(x))

    for :math:`-\pi \le x \le \pi`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zcosine_gen._shape_info  r   r1   c                 P    dt           j        z  dt          j        |          z   z  S )Nr   r   rC   r   r  r   s     r/   re   zcosine_gen._pdf  s    RU{AbfQiiK((r1   c                 r    t          j        |          }t          |dk    |fd t           j                   S )Nr  c                 n    t          j        |           t          j        dt           j        z            z
  S r	  )rC   r  r   r   r  s    r/   r  z$cosine_gen._logpdf.<locals>.<lambda>  s!    BHQKK"&25//$A r1   r  )rC   r  r   r\   r  s      r/   r   zcosine_gen._logpdf  s=    F1II!r'A4AA%'VG- - - 	-r1   c                 *    t          j        |          S rA   ra   _cosine_cdfr   s     r/   rh   zcosine_gen._cdf  s    q!!!r1   c                 ,    t          j        |           S rA   r  r   s     r/   rl   zcosine_gen._sf  s    r"""r1   c                 *    t          j        |          S rA   ra   _cosine_invcdfr:   r}  s     r/   rq   zcosine_gen._ppf  s    !!$$$r1   c                 ,    t          j        |           S rA   r  r  s     r/   rt   zcosine_gen._isf  s    "1%%%%r1   c                     dt           j        t           j        z  dz  dz
  ddt           j        dz  dz
  z  dt           j        t           j        z  dz
  d	z  z  z  fS )
Nr{   r  r         r  Z         @r  rH   r   r]   s    r/   r   zcosine_gen._stats  sN    BE"%KOC'dBE1HRK.@#ruRU{ST}WXFXBX.YYYr1   c                 J    t          j        dt           j        z            dz
  S )Nr  r|   r   r]   s    r/   r   zcosine_gen._entropy  s    vags""r1   Nrv   rw   rx   ry   r^   re   r   rh   rl   rq   rt   r   r   rz   r1   r/   r  r    s         (  ) ) )- - -" " "# # #% % %& & &Z Z Z# # # # #r1   r  cosinec                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )
dgamma_gena  A double gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dgamma` is:

    .. math::

        f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)

    for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `dgamma` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r   r[   r]   s    r/   r^   zdgamma_gen._shape_info  r   r1   Nc                     |                     |          }t                              |||          }|t          j        |dk    dd          z  S Nr   r  r   r   r  )uniformr  r  rC   r  )r:   r~   r   r   ugms         r/   r   zdgamma_gen._rvs  sL      d ++YYqt,Y??BHQ#Xq"----r1   c                     t          |          }ddt          j        |          z  z  ||dz
  z  z  t          j        |           z  S r   )absrj   r  rC   r   r:   rd   r~   axs       r/   re   zdgamma_gen._pdf  s@    VVAbhqkkM"2#;.<<r1   c                     t          |          }t          j        |dz
  |          |z
  t          j        d          z
  t          j        |          z
  S r   )r  rj   rf  rC   r   rW  r  s       r/   r   zdgamma_gen._logpdf  sB    VVxC$$r)BF1II5
1EEr1   c                     dt          j        |t          |                    z  }t          j        |dk    d|z   d|z
            S Nr   r   rj   r\  r  rC   r  r:   rd   r~   r  s       r/   rh   zdgamma_gen._cdf  s>    "+aQ(((xAsSy#)444r1   c                     dt          j        |t          |                    z  }t          j        |dk    d|z
  d|z             S r  r  r  s       r/   rl   zdgamma_gen._sf  s>    "+aQ(((xAs3wC000r1   c                     t          j        |dt          d|z  dz
            z
            }t          j        |dk    ||           S Nr   rH   r   )rj   rg  r  rC   r  )r:   rp   r~   r  s       r/   rq   zdgamma_gen._ppf  s?    oa3qs1u::..xCsd+++r1   c                 <    ||dz   z  }d|d|dz   |dz   z  |z  dz
  fS )Nr|   r{   r   r  rz   )r:   r~   r6  s      r/   r   zdgamma_gen._stats
  s4    3iCququoc1#555r1   r   rv   rw   rx   ry   r^   r   re   r   rh   rl   rq   r   rz   r1   r/   r  r    s         ,E E E. . . .
= = =
F F F5 5 51 1 1, , ,6 6 6 6 6r1   r  dgammac                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )dweibull_genav  A double Weibull continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dweibull` is given by

    .. math::

        f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)

    for a real number :math:`x` and :math:`c > 0`.

    `dweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zdweibull_gen._shape_info(  r   r1   Nc                     |                     |          }t                              |||          }|t          j        |dk    dd          z  S r  )r  weibull_minr  rC   r  )r:   r  r   r   r  ws         r/   r   zdweibull_gen._rvs+  sL      d ++OOAD|ODDBHQ#Xq"--..r1   c                 r    t          |          }|dz  ||dz
  z  z  t          j        ||z             z  }|S Nr   r|   )r  rC   r   )r:   rd   r  r  Pxs        r/   re   zdweibull_gen._pdf0  s;    VVWrAcE{"RVRUF^^3	r1   c                     t          |          }t          j        |          t          j        d          z
  t          j        |dz
  |          z   ||z  z
  S r  )r  rC   r   rj   rf  )r:   rd   r  r  s       r/   r   zdweibull_gen._logpdf6  sF    VVvayy26#;;&!c'2)>)>>QFFr1   c                     dt          j        t          |          |z             z  }t          j        |dk    d|z
  |          S )Nr   r   r   )rC   r   r  r  )r:   rd   r  Cx1s       r/   rh   zdweibull_gen._cdf:  s>    BFCFFAI:&&&xAq3w,,,r1   c                     dt          j        |dk    |d|z
            z  }t          j        t          j        |           d|z            }t          j        |dk    ||           S )Nr   r   r|   )rC   r  rj  r   )r:   rp   r  r  s       r/   rq   zdweibull_gen._ppf>  s[    28AHaa000hs|S1W--xCsd+++r1   c                 N    d|dz  z
  t          j        dd|z  |z  z             z  S )Nr   rH   r|   rj   r  r'  s      r/   r   zdweibull_gen._munpC  s,    QUrxcAgk(9::::r1   c                     dS N)r   Nr   Nrz   r)  s     r/   r   zdweibull_gen._statsI      r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rq   r   r   rz   r1   r/   r  r    s         *E E E/ / / /
  G G G- - -, , ,
; ; ;         r1   r  dweibullc                       e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Ze eed          d                         ZdS )	expon_genaE  An exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `expon` is:

    .. math::

        f(x) = \exp(-x)

    for :math:`x \ge 0`.

    %(after_notes)s

    A common parameterization for `expon` is in terms of the rate parameter
    ``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
    parameterization corresponds to using ``scale = 1 / lambda``.

    The exponential distribution is a special case of the gamma
    distributions, with gamma shape parameter ``a = 1``.

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zexpon_gen._shape_infok  r   r1   Nc                 ,    |                     |          S rA   )standard_exponentialr   s      r/   r   zexpon_gen._rvsn  s    00666r1   c                 ,    t          j        |           S rA   rC   r   r   s     r/   re   zexpon_gen._pdfq  s    vqbzzr1   c                     | S rA   rz   r   s     r/   r   zexpon_gen._logpdfu  	    r	r1   c                 .    t          j        |            S rA   rj   r  r   s     r/   rh   zexpon_gen._cdfx      !}r1   c                 .    t          j        |            S rA   r  r   s     r/   rq   zexpon_gen._ppf{  r  r1   c                 ,    t          j        |           S rA   r  r   s     r/   rl   zexpon_gen._sf~  s    vqbzzr1   c                     | S rA   rz   r   s     r/   r   zexpon_gen._logsf  r  r1   c                 ,    t          j        |           S rA   r#  r   s     r/   rt   zexpon_gen._isf      q		zr1   c                     dS )N)r|   r|   r         @rz   r]   s    r/   r   zexpon_gen._stats  r   r1   c                     dS r  rz   r]   s    r/   r   zexpon_gen._entropy      sr1   z        When `method='MLE'`,
        this function uses explicit formulas for the maximum likelihood
        estimation of the exponential distribution parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are
        ignored.

r   c                 b   t          |          dk    rt          d          |                    dd           }|                    dd           }t          |           ||t	          d          t          j        |          }t          j        |                                          st	          d          |	                                }||}n$|}||k     rt          d|t
          j                  ||                                |z
  }n|}t          |          t          |          fS )	Nr   Too many arguments.r   r   r   r   exponr  )r  r-   r,   r0   r   rC   r   r   r   minr<  r\   r   float)	r:   r   r;   r.   r   r   data_minr'   r(   s	            r/   r9   zexpon_gen.fit  s,    t99q==1222xx%%(D))$T*** 2 ) * * * z${4  $$&& 	ECDDD88::<CCC#~~"7$bfEEEE>IIKK#%EEE Szz5<<''r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rq   rl   r   rt   r   r   r>   r	   r   r9   rz   r1   r/   r  r  P  s	        4  7 7 7 7              " " "    6   &( &(  _&( &( &(r1   r  r  c                   >    e Zd ZdZd Zd
dZd Zd Zd Zd Z	d	 Z
dS )exponnorm_gena  An exponentially modified Normal continuous random variable.

    Also known as the exponentially modified Gaussian distribution [1]_.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponnorm` is:

    .. math::

        f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
                  \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)

    where :math:`x` is a real number and :math:`K > 0`.

    It can be thought of as the sum of a standard normal random variable
    and an independent exponentially distributed random variable with rate
    ``1/K``.

    %(after_notes)s

    An alternative parameterization of this distribution (for example, in
    the Wikpedia article [1]_) involves three parameters, :math:`\mu`,
    :math:`\lambda` and :math:`\sigma`.

    In the present parameterization this corresponds to having ``loc`` and
    ``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
    shape parameter :math:`K = 1/(\sigma\lambda)`.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Exponentially modified Gaussian distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution

    %(example)s

    c                 @    t          dddt          j        fd          gS )NKFr   r   r[   r]   s    r/   r^   zexponnorm_gen._shape_info  r   r1   Nc                 f    |                     |          |z  }|                    |          }||z   S rA   )r  r   )r:   r  r   r   expvalgvals         r/   r   zexponnorm_gen._rvs  s7    224881<++D11}r1   c                 R    t          j        |                     ||                    S rA   r  )r:   rd   r  s      r/   re   zexponnorm_gen._pdf       vdll1a(()))r1   c                 v    d|z  }|d|z  |z
  z  }|t          ||z
            z   t          j        |          z
  S Nr|   r   r   rC   r   )r:   rd   r  invKexpargs        r/   r   zexponnorm_gen._logpdf  sA    Qwta(QX...::r1   c                     d|z  }|d|z  |z
  z  }|t          ||z
            z   }t          |          t          j        |          z
  S r  r   r   rC   r   r:   rd   r  r  r  logprods         r/   rh   zexponnorm_gen._cdf  sL    Qwta(<D111||bfWoo--r1   c                     d|z  }|d|z  |z
  z  }|t          ||z
            z   }t          |           t          j        |          z   S r  r  r  s         r/   rl   zexponnorm_gen._sf   sN    Qwta(<D111!}}rvg..r1   c                 Z    ||z  }d|z   }d|dz  z  |dz  z  }d|z  |z  |dz  z  }||||fS )Nr|   rH   r  r3  r  r  rz   )r:   r  K2opK2skwkrts         r/   r   zexponnorm_gen._stats  sO    URx!Q$h%BhmdRj($S  r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   r   rz   r1   r/   r  r    s        ( (RE E E   
* * *; ; ;
. . ./ / /! ! ! ! !r1   r  	exponnormc                   0    e Zd ZdZd Zd Zd Zd Zd ZdS )exponweib_gena  An exponentiated Weibull continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, numpy.random.Generator.weibull

    Notes
    -----
    The probability density function for `exponweib` is:

    .. math::

        f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}

    and its cumulative distribution function is:

    .. math::

        F(x, a, c) = [1-\exp(-x^c)]^a

    for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.

    `exponweib` takes :math:`a` and :math:`c` as shape parameters:

    * :math:`a` is the exponentiation parameter,
      with the special case :math:`a=1` corresponding to the
      (non-exponentiated) Weibull distribution `weibull_min`.
    * :math:`c` is the shape parameter of the non-exponentiated Weibull law.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS Nr~   Fr   r   r  r[   r:   rZ  r  s      r/   r^   zexponweib_gen._shape_info:  r\  r1   c                 T    t          j        |                     |||                    S rA   r  r:   rd   r~   r  s       r/   re   zexponweib_gen._pdf?  $     vdll1a++,,,r1   c                     ||z   }t          j        |           }t          j        |          t          j        |          z   t          j        |dz
  |          z   |z   t          j        |dz
  |          z   }|S r  )rj   r  rC   r   rf  )r:   rd   r~   r  negxcexm1clogps          r/   r   zexponweib_gen._logpdfD  sq    A% q		BF1II%S%(@(@@S!,,-r1   c                 >    t          j        ||z              }||z  S rA   r  )r:   rd   r~   r  r  s        r/   rh   zexponweib_gen._cdfK  s!    1a4% axr1   c                 j    t          j        |d|z  z              t          j        d|z            z  S r  )rj   r  rC   r   )r:   rp   r~   r  s       r/   rq   zexponweib_gen._ppfO  s2    1s1u:+&&&CE):):::r1   N	rv   rw   rx   ry   r^   re   r   rh   rq   rz   r1   r/   r  r    sj        ' 'P  
- - -
    ; ; ; ; ;r1   r  	exponweibc                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
exponpow_gena  An exponential power continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponpow` is:

    .. math::

        f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))

    for :math:`x \ge 0`, :math:`b > 0`.  Note that this is a different
    distribution from the exponential power distribution that is also known
    under the names "generalized normal" or "generalized Gaussian".

    `exponpow` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    References
    ----------
    http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf

    %(example)s

    c                 @    t          dddt          j        fd          gS Nr   Fr   r   r[   r]   s    r/   r^   zexponpow_gen._shape_infor  r   r1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   r   s      r/   re   zexponpow_gen._pdfu       vdll1a(()))r1   c                     ||z  }dt          j        |          z   t          j        |dz
  |          z   |z   t          j        |          z
  }|S Nr   r|   )rC   r   rj   rf  r   )r:   rd   r   xbfs        r/   r   zexponpow_gen._logpdfy  sH    Tq		MBHQWa00025r

Br1   c                 X    t          j        t          j        ||z                        S rA   r  r  s      r/   rh   zexponpow_gen._cdf~  s#    "(1a4..))))r1   c                 V    t          j        t          j        ||z                       S rA   rC   r   rj   r  r  s      r/   rl   zexponpow_gen._sf  s     vrx1~~o&&&r1   c                 \    t          j        t          j        |                     d|z  z  S r  rj   r  rC   r   r  s      r/   rt   zexponpow_gen._isf  s%    "&))$$1--r1   c                 t    t          t          j        t          j        |                      d|z            S r  powrj   r  r:   rp   r   s      r/   rq   zexponpow_gen._ppf  s,    28RXqb\\M**CE222r1   N)rv   rw   rx   ry   r^   re   r   rh   rl   rt   rq   rz   r1   r/   r
  r
  V  s         6E E E* * *  
* * *' ' '. . .3 3 3 3 3r1   r
  exponpowc                   X    e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd ZdS )fatiguelife_gena0  A fatigue-life (Birnbaum-Saunders) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `fatiguelife` is:

    .. math::

        f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})

    for :math:`x >= 0` and :math:`c > 0`.

    `fatiguelife` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] "Birnbaum-Saunders distribution",
           https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zfatiguelife_gen._shape_info  r   r1   Nc                     |                     |          }d|z  |z  }||z  }dd|z  z   d|z  t          j        d|z             z  z   }|S )Nr   r|   rH   r   )r   rC   r   )r:   r  r   r   zrd   x2ts           r/   r   zfatiguelife_gen._rvs  sW    ((..E!GqS!B$J1RWQV__,,r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zfatiguelife_gen._pdf  s"     vdll1a(()))r1   c                    t          j        |dz             |dz
  dz  d|z  |dz  z  z  z
  t          j        d|z            z
  dt          j        dt           j        z            dt          j        |          z  z   z  z
  S )Nr   rH   r   r   r  r   r  s      r/   r   zfatiguelife_gen._logpdf  sq    qsqsQh#a%1*55qsCRVAbeG__q{234 	5r1   c                     t          d|z  t          j        |          dt          j        |          z  z
  z            S r  )r   rC   r   r  s      r/   rh   zfatiguelife_gen._cdf  s2    qBGAJJRWQZZ$?@AAAr1   c                 v    |t          j        |          z  }d|t          j        |dz  dz             z   dz  z  S N      ?rH   r  rj   r   rC   r   r:   rp   r  tmps       r/   rq   zfatiguelife_gen._ppf  s:    msRWS!VaZ0001444r1   c                     t          d|z  t          j        |          dt          j        |          z  z
  z            S r  )r   rC   r   r  s      r/   rl   zfatiguelife_gen._sf  s2    a271::BGAJJ#>?@@@r1   c                 x    | t          j        |          z  }d|t          j        |dz  dz             z   dz  z  S r)  r+  r,  s       r/   rt   zfatiguelife_gen._isf  s<    b!nsRWS!VaZ0001444r1   c                     ||z  }|dz  dz   }d|z  dz   }||z  dz  }d|z  d|z  dz   z  t          j        |d          z  }d	|z  d
|z  dz   z  |dz  z  }||||fS )Nr   r|   r  r  r     r  ri  r  ]   g      D@rC   rj  )r:   r  c2r5  denr6  r7  r8  s           r/   r   zfatiguelife_gen._stats  s     qS#X^BhnfslUbeck"RXc3%7%77Vr"ut|$sCx/3Br1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   rh   rq   rl   rt   r   rz   r1   r/   r  r    s         4 "4ME E E   * * *
5 5 5B B B5 5 5A A A5 5 5    r1   r  fatiguelifec                   8    e Zd ZdZd Zd Zd	dZd Zd Zd Z	dS )
foldcauchy_genao  A folded Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldcauchy` is:

    .. math::

        f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(example)s

    c                     |dk    S Nr   rz   r)  s     r/   rV   zfoldcauchy_gen._argcheck      Avr1   c                 @    t          dddt          j        fd          gS Nr  Fr   rZ   r[   r]   s    r/   r^   zfoldcauchy_gen._shape_info      326{MBBCCr1   Nc                 V    t          t                              |||                    S )Nr'   r   r   )r  rJ  r  r:   r  r   r   s       r/   r   zfoldcauchy_gen._rvs  s0    6::!$+7  9 9 : : 	:r1   c                 \    dt           j        z  dd||z
  dz  z   z  dd||z   dz  z   z  z   z  S Nr|   r   rH   r   r  s      r/   re   zfoldcauchy_gen._pdf  s:    25y#q!A#z*S!QqS1H*-==>>r1   c                     dt           j        z  t          j        ||z
            t          j        ||z             z   z  S r  r3  r  s      r/   rh   zfoldcauchy_gen._cdf   s0    25y")AaC..29QqS>>9::r1   c                 ^    t           j        t           j        t           j        t           j        fS rA   r
  r)  s     r/   r   zfoldcauchy_gen._stats  r=  r1   r   
rv   rw   rx   ry   rV   r^   r   re   rh   r   rz   r1   r/   r8  r8    s         &  D D D: : : :? ? ?; ; ;. . . . .r1   r8  
foldcauchyc                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )f_genaE  An F continuous random variable.

    For the noncentral F distribution, see `ncf`.

    %(before_notes)s

    See Also
    --------
    ncf

    Notes
    -----
    The probability density function for `f` is:

    .. math::

        f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
                                {(df_2+df_1 x)^{(df_1+df_2)/2}
                                 B(df_1/2, df_2/2)}

    for :math:`x > 0` and parameters :math:`df_1, df_2 > 0` .

    `f` takes ``dfn`` and ``dfd`` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )NdfnFr   r   dfdr[   )r:   idfnidfds      r/   r^   zf_gen._shape_info(  s>    %BF^DD%BF^DDd|r1   Nc                 0    |                     |||          S rA   )r  )r:   rK  rL  r   r   s        r/   r   z
f_gen._rvs-  s    ~~c3---r1   c                 T    t          j        |                     |||                    S rA   r  r:   rd   rK  rL  s       r/   re   z
f_gen._pdf0  s$     vdll1c3//000r1   c                 <   d|z  }d|z  }|dz  t          j        |          z  |dz  t          j        |          z  z   t          j        |dz  dz
  |          z   ||z   dz  t          j        |||z  z             z  t          j        |dz  |dz            z   z
  }|S Nr|   rH   r   )rC   r   rj   rf  rg  )r:   rd   rK  rL  rU   mrh  s          r/   r   zf_gen._logpdf6  s    #I#IsRVAYY1rvayy028AaC!GQ3G3GGaC7bfQ1Woo-	!A#qs0C0CCE
r1   c                 .    t          j        |||          S rA   )rj   fdtrrQ  s       r/   rh   z
f_gen._cdf=  s    wsC###r1   c                 .    t          j        |||          S rA   )rj   fdtrcrQ  s       r/   rl   z	f_gen._sf@      xS!$$$r1   c                 .    t          j        |||          S rA   )rj   fdtri)r:   rp   rK  rL  s       r/   rq   z
f_gen._ppfC  rY  r1   c                    d|z  d|z  }}|dz
  |dz
  |dz
  |dz
  f\  }}}}t          |dk    ||fd t          j                  }	t          |dk    ||||fd	 t          j                  }
t          |d
k    ||||fd t          j                  }|t          j        d          z  }t          |dk    |||fd t          j                  }|dz  }|	|
||fS )Nr|   r   r  r         @rH   c                     | |z  S rA   rz   )v2v2_2s     r/   r  zf_gen._stats.<locals>.<lambda>L  s
    R$Y r1   r  c                 6    d|z  |z  | |z   z  | |dz  z  |z  z  S r	  rz   )v1r_  r`  v2_4s       r/   r  zf_gen._stats.<locals>.<lambda>Q  s-    FRK29%dAg)<= r1   r  c                 T    d| z  |z   |z  t          j        || | |z   z  z            z  S r	  r  )rb  r`  rc  v2_6s       r/   r  zf_gen._stats.<locals>.<lambda>W  s4    Vd]d"RWTR295E-F%G%GG r1   r  c                     d| | z  |z  z   |z  S )Nr  rz   )r7  re  v2_8s      r/   r  zf_gen._stats.<locals>.<lambda>^  s    AR$$6$#> r1   ri  )r   rC   r\   r  r   )r:   rK  rL  rb  r_  r`  rc  re  rg  r5  r6  r7  r8  s                r/   r   zf_gen._statsF  s   c28B!#b"r'27BG!CdD$FRJ&&F 
 FRT4(> >F	  FRtT*H HF	 
 	bgbkkFRt$>>F  	g3Br1   r   r  rz   r1   r/   rI  rI  
  s         :  
. . . .1 1 1  $ $ $% % %% % %    r1   rI  r  c                   8    e Zd ZdZd Zd Zd	dZd Zd Zd Z	dS )
foldnorm_genaz  A folded normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldnorm` is:

    .. math::

        f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldnorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                     |dk    S r:  rz   r)  s     r/   rV   zfoldnorm_gen._argcheck  r;  r1   c                 @    t          dddt          j        fd          gS r=  r[   r]   s    r/   r^   zfoldnorm_gen._shape_info  r>  r1   Nc                 L    t          |                    |          |z             S rA   r  r   rA  s       r/   r   zfoldnorm_gen._rvs  s#    <//559:::r1   c                 L    t          ||z             t          ||z
            z   S rA   r   r  s      r/   re   zfoldnorm_gen._pdf  s#    Q)AaC..00r1   c                 R    t          ||z
            t          ||z             z   dz
  S r  r   r  s      r/   rh   zfoldnorm_gen._cdf  s&    1~~	!A#.44r1   c                    ||z  }t          j        d|z            t          j        dt           j        z            z  }d|z  |t	          j        |t          j        d          z            z  z   }|dz   ||z  z
  }d||z  |z  ||z  z
  |z
  z  }|t          j        |d          z  }||dz   z  dz   d|z  |z  z   }|d|d	z
  z  d	|dz  z  z
  |dz  z  z  }||dz  z  d	z
  }||||fS )
N      r   rH   r   ri  r  r  r]  r  )rC   r   r   r   rj   erfrj  )r:   r  r4  expfacr5  r6  r7  r8  s           r/   r   zfoldnorm_gen._stats  s
    qSR272be8#4#44YRVAbgajjL11111fr"un2b58be#f,-
bhsC   27^a"V)B,.
rR"W~RU
*b!e33#s(]R3Br1   r   rF  rz   r1   r/   ri  ri  p  s         *  D D D; ; ; ;1 1 15 5 5    r1   ri  foldnormc                   ~     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Z eed           fd            Z xZS )weibull_min_genaF  Weibull minimum continuous random variable.

    The Weibull Minimum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is also often simply called the Weibull
    distribution. It arises as the limiting distribution of the rescaled
    minimum of iid random variables.

    %(before_notes)s

    See Also
    --------
    weibull_max, numpy.random.Generator.weibull, exponweib

    Notes
    -----
    The probability density function for `weibull_min` is:

    .. math::

        f(x, c) = c x^{c-1} \exp(-x^c)

    for :math:`x > 0`, :math:`c > 0`.

    `weibull_min` takes ``c`` as a shape parameter for :math:`c`.
    (named :math:`k` in Wikipedia article and :math:`a` in
    ``numpy.random.weibull``).  Special shape values are :math:`c=1` and
    :math:`c=2` where Weibull distribution reduces to the `expon` and
    `rayleigh` distributions respectively.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zweibull_min_gen._shape_info  r   r1   c                 v    |t          ||dz
            z  t          j        t          ||                     z  S rQ   r  rC   r   r  s      r/   re   zweibull_min_gen._pdf  s1    Q!}RVSAYYJ////r1   c                 ~    t          j        |          t          j        |dz
  |          z   t	          ||          z
  S rQ   rC   r   rj   rf  r  r  s      r/   r   zweibull_min_gen._logpdf  s2    vayy28AE1---Aq		99r1   c                 J    t          j        t          ||                      S rA   rj   r  r  r  s      r/   rh   zweibull_min_gen._cdf  s    #a))$$$$r1   c                 H    t          j        t          ||                     S rA   rC   r   r  r  s      r/   rl   zweibull_min_gen._sf  s    vs1ayyj!!!r1   c                 $    t          ||           S rA   r  r  s      r/   r   zweibull_min_gen._logsf  s    Aq		zr1   c                 P    t          t          j        |            d|z            S r  r  r  s      r/   rq   zweibull_min_gen._ppf  s"    BHaRLL=#a%(((r1   c                 <    t          j        d|dz  |z  z             S r  r  r'  s      r/   r   zweibull_min_gen._munp  s    xAcE!G$$$r1   c                 X    t            |z  t          j        |          z
  t           z   dz   S rQ   r   rC   r   r)  s     r/   r   zweibull_min_gen._entropy  %    w{RVAYY&/!33r1   a          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        

r   c           	         |                     dd          r t                      j        |g|R i |S t          | |||          \  }}}}|                    dd                                          }d t          j        |          d} |          }	|	k     r'|dk    r!||s t                      j        |g|R i |S |dk    rd\  }
}}nEt          |          r|d	         nd }
|                     d
d           }|                     dd           }| |
t          fdd|gd          j
        }
n||}
|d|bt          j        |          }t          j        |t          j        dd|
z  z             t          j        dd|
z  z             dz  z
  z            }n||}|7|5t          j        |          }||t          j        dd|
z  z             z  z
  }n||}|dk    r|
||fS  t                      j        ||
f||d|S )NsuperfitFr*   r4   c                     t          j        dd| z  z             }t          j        dd| z  z             }t          j        dd| z  z             }d|dz  z  d|z  |z  z
  |z   }||dz  z
  dz  }||z  S )Nr   rH   r  ri  r  )r  gamma1gamma2gamma3numr5  s         r/   skewz!weibull_min_gen.fit.<locals>.skew	  s|    Xa!e__FXa!e__FXa!e__Ffai-!F(6/1F:CFAI%-Cs7Nr1   g     @mmNNNr   r'   r(   c                       |           z
  S rA   rz   )r  r  r  s    r/   r  z%weibull_min_gen.fit.<locals>.<lambda>'	  s    dd1ggk r1   g{Gz?bisect)bracketr*   r   rH   r'   r(   )r,   r7   r9   _check_fit_input_parametersr5   r6   statsr  r  r$   rootrC   r  r   rj   r  r   )r:   r   r;   r.   fcr   r   r*   max_cs_minr  r'   r(   vrT  r  r  r  s                  @@r/   r9   zweibull_min_gen.fit  sz    88J&& 	4577;t3d333d333 "=T4=A4"I "Ib$(E**0022	 	 	 JtUu994BJtJ577;t3d333d333 T>>,MAsEEt99.Q$A((5$''CHHWd++E:!) 11111D%=#+- - --1 A^A>emtAGA!AaC%28AacE??A3E!EFGGEEE<CKAeBHQ1W----CCCT>>c5=  577;tQECuEEEEEr1   )rv   rw   rx   ry   r^   re   r   rh   rl   r   rq   r   r   r   r   r9   r  r  s   @r/   rv  rv    s       ' 'PE E E0 0 0: : :% % %" " "  ) ) )% % %4 4 4 } 5   CF CF CF CF CF CF CF CF CFr1   rv  r  c                   j     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd Z xZS )truncweibull_min_gena9  A doubly truncated Weibull minimum continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, truncexpon

    Notes
    -----
    The probability density function for `truncweibull_min` is:

    .. math::

        f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)}

    for :math:`a < x <= b`, :math:`0 \le a < b` and :math:`c > 0`.

    `truncweibull_min` takes :math:`a`, :math:`b`, and :math:`c` as shape
    parameters.

    Notice that the truncation values, :math:`a` and :math:`b`, are defined in
    standardized form:

    .. math::

        a = (u_l - loc)/scale
        b = (u_r - loc)/scale

    where :math:`u_l` and :math:`u_r` are the specific left and right
    truncation values, respectively. In other words, the support of the
    distribution becomes :math:`(a*scale + loc) < x <= (b*scale + loc)` when
    :math:`loc` and/or :math:`scale` are provided.

    %(after_notes)s

    References
    ----------

    .. [1] Rinne, H. "The Weibull Distribution: A Handbook". CRC Press (2009).

    %(example)s

    c                 *    |dk    ||k    z  |dk    z  S Nr{   rz   r:   r  r~   r   s       r/   rV   ztruncweibull_min_gen._argcheckp	  s    RAE"a"f--r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }t          dddt          j        fd          }|||gS )Nr  Fr   r   r~   rZ   r   r[   )r:   r  rZ  r[  s       r/   r^   z truncweibull_min_gen._shape_infos	  sY    UQK@@UQK??UQK@@B|r1   c                 J    t                                          |d          S )N)r   r   r   r  r7   r  r:   r   r  s     r/   r  ztruncweibull_min_gen._fitstarty	  s     ww  I 666r1   c                 
    ||fS rA   rz   r  s       r/   r   z!truncweibull_min_gen._get_support}	      !tr1   c                 
   t          j        t          ||                     t          j        t          ||                     z
  }|t          ||dz
            z  t          j        t          ||                     z  |z  S rQ   r  )r:   rd   r  r~   r   denums         r/   re   ztruncweibull_min_gen._pdf	  sh    Q
##bfc!QiiZ&8&88C1Q3KK"&#a))"4"44==r1   c           	      6   t          j        t          j        t          ||                     t          j        t          ||                     z
            }t          j        |          t	          j        |dz
  |          z   t          ||          z
  |z
  S rQ   )rC   r   r   r  rj   rf  )r:   rd   r  r~   r   logdenums         r/   r   ztruncweibull_min_gen._logpdf	  ss    6"&#a)),,rvs1ayyj/A/AABBvayy28AE1---Aq		9HDDr1   c                 (   t          j        t          ||                     t          j        t          ||                     z
  }t          j        t          ||                     t          j        t          ||                     z
  }||z  S rA   r  r:   rd   r  r~   r   r  r  s          r/   rh   ztruncweibull_min_gen._cdf	  p    vs1ayyj!!BFC1II:$6$66Q
##bfc!QiiZ&8&88U{r1   c           	      p   t          j        t          j        t          ||                     t          j        t          ||                     z
            }t          j        t          j        t          ||                     t          j        t          ||                     z
            }||z
  S rA   rC   r   r   r  r:   rd   r  r~   r   lognumr  s          r/   r   ztruncweibull_min_gen._logcdf	      Aq		z**RVSAYYJ-?-??@@6"&#a)),,rvs1ayyj/A/AABB  r1   c                 (   t          j        t          ||                     t          j        t          ||                     z
  }t          j        t          ||                     t          j        t          ||                     z
  }||z  S rA   r  r  s          r/   rl   ztruncweibull_min_gen._sf	  r  r1   c           	      p   t          j        t          j        t          ||                     t          j        t          ||                     z
            }t          j        t          j        t          ||                     t          j        t          ||                     z
            }||z
  S rA   r  r  s          r/   r   ztruncweibull_min_gen._logsf	  r  r1   c                     t          t          j        d|z
  t          j        t          ||                     z  |t          j        t          ||                     z  z              d|z            S rQ   r  rC   r   r   r:   rp   r  r~   r   s        r/   rt   ztruncweibull_min_gen._isf	  e    VQUbfc!QiiZ0001rvs1ayyj7I7I3IIJJJAaC  	r1   c                     t          t          j        d|z
  t          j        t          ||                     z  |t          j        t          ||                     z  z              d|z            S rQ   r  r  s        r/   rq   ztruncweibull_min_gen._ppf	  r  r1   c           	      v   t          j        ||z  dz             t          j        ||z  dz   t          ||                    t          j        ||z  dz   t          ||                    z
  z  }t	          j        t          ||                     t	          j        t          ||                     z
  }||z  S r  )rj   r  r\  r  rC   r   )r:   rU   r  r~   r   	gamma_funr  s          r/   r   ztruncweibull_min_gen._munp	  s    HQqS2X&&K!b#a)),,r{1Q38SAYY/O/OO	 Q
##bfc!QiiZ&8&885  r1   )rv   rw   rx   ry   rV   r^   r  r   re   r   rh   r   rl   r   rt   rq   r   r  r  s   @r/   r  r  C	  s        + +X. . .  7 7 7 7 7  > > >E E E  
! ! !
  
! ! !
  
  
! ! ! ! ! ! !r1   r  truncweibull_minc                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZdS )weibull_max_gena0  Weibull maximum continuous random variable.

    The Weibull Maximum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is the limiting distribution of rescaled
    maximum of iid random variables. This is the distribution of -X
    if X is from the `weibull_min` function.

    %(before_notes)s

    See Also
    --------
    weibull_min

    Notes
    -----
    The probability density function for `weibull_max` is:

    .. math::

        f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)

    for :math:`x < 0`, :math:`c > 0`.

    `weibull_max` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zweibull_max_gen._shape_info	  r   r1   c                 z    |t          | |dz
            z  t          j        t          | |                     z  S rQ   ry  r  s      r/   re   zweibull_max_gen._pdf	  s5    aR1~bfc1"ajj[1111r1   c                     t          j        |          t          j        |dz
  |           z   t	          | |          z
  S rQ   r{  r  s      r/   r   zweibull_max_gen._logpdf	  s6    vayy28AaC!,,,sA2qzz99r1   c                 J    t          j        t          | |                     S rA   r  r  s      r/   rh   zweibull_max_gen._cdf	  s    vsA2qzzk"""r1   c                 &    t          | |           S rA   r  r  s      r/   r   zweibull_max_gen._logcdf	  s    QB

{r1   c                 L    t          j        t          | |                      S rA   r}  r  s      r/   rl   zweibull_max_gen._sf	  s!    #qb!**%%%%r1   c                 P    t          t          j        |           d|z             S r  )r  rC   r   r  s      r/   rq   zweibull_max_gen._ppf	  s#    RVAYYJA&&&&r1   c                 t    t          j        d|dz  |z  z             }t          |          dz  rd}nd}||z  S )Nr|   rH   r  r   )rj   r  int)r:   rU   r  valsgns        r/   r   zweibull_max_gen._munp	  sE    hs1S57{##q66A: 	CCCSyr1   c                 X    t            |z  t          j        |          z
  t           z   dz   S rQ   r  r)  s     r/   r   zweibull_max_gen._entropy	  r  r1   N)rv   rw   rx   ry   r^   re   r   rh   r   rl   rq   r   r   rz   r1   r/   r  r  	  s        # #HE E E2 2 2: : :# # #  & & &' ' '  4 4 4 4 4r1   r  weibull_max)r   r   c                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	genlogistic_gena  A generalized logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genlogistic` is:

    .. math::

        f(x, c) = c \frac{\exp(-x)}
                         {(1 + \exp(-x))^{c+1}}

    for :math:`x >= 0`, :math:`c > 0`.

    `genlogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zgenlogistic_gen._shape_info
  r   r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zgenlogistic_gen._pdf
  r  r1   c                     |dz
   |dk     z  dz
  }t          j        |          }t          j        |          ||z  z   |dz   t          j        t          j        |                     z  z
  S Nr   r   )rC   r  r   rj   r  r   )r:   rd   r  multabsxs        r/   r   zgenlogistic_gen._logpdf
  sc     Qx1q5!A%vayyvayy49$!rxu/F/F'FFFr1   c                 >    dt          j        |           z   | z  }|S rQ   r  )r:   rd   r  Cxs       r/   rh   zgenlogistic_gen._cdf!
  s!    r

lqb!	r1   c                 X    t          j        t          |d|z            dz
             }|S r  rC   r   r  )r:   rp   r  valss       r/   rq   zgenlogistic_gen._ppf%
  s*    s1d1f~~a'(((r1   c                    t           t          j        |          z   }t          j        t          j        z  dz  t          j        d|          z   }dt          j        d|          z  dt          z  z   }|t          j        |d          z  }t          j        dz  dz  dt          j        d|          z  z   }||d	z  z  }||||fS )
Nr  rH   r  r  ri  r        .@r  r   )r   rj   rL  rC   r   zetar    rj  r:   r  r5  r6  r7  r8  s         r/   r   zgenlogistic_gen._stats)
  s    bfQiieBEk#o1-1&(
bhsC   UAXd]Qrwq!}}_,
c3h3Br1   N)
rv   rw   rx   ry   r^   re   r   rh   rq   r   rz   r1   r/   r  r  	  s~         ,E E E* * *G G G        r1   r  genlogisticc                   b    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd ZddZd Zd ZdS )genpareto_gena  A generalized Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genpareto` is:

    .. math::

        f(x, c) = (1 + c x)^{-1 - 1/c}

    defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
    :math:`0 \le x \le -1/c` if :math:`c < 0`.

    `genpareto` takes ``c`` as a shape parameter for :math:`c`.

    For :math:`c=0`, `genpareto` reduces to the exponential
    distribution, `expon`:

    .. math::

        f(x, 0) = \exp(-x)

    For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:

    .. math::

        f(x, -1) = 1

    %(after_notes)s

    %(example)s

    c                 *    t          j        |          S rA   rC   r   r)  s     r/   rV   zgenpareto_gen._argcheckZ
      {1~~r1   c                 V    t          ddt          j         t          j        fd          gS Nr  Fr   r[   r]   s    r/   r^   zgenpareto_gen._shape_info]
  $    3'8.IIJJr1   c                     t          j        |          }t          |dk     |fd t           j                  }t          j        |dk    | j        | j                  }||fS )Nr   c                     d| z  S Nr  rz   r  s    r/   r  z,genpareto_gen._get_support.<locals>.<lambda>c
  s
    q r1   )rC   r   r   r\   r  r~   )r:   r  r   r~   s       r/   r   zgenpareto_gen._get_support`
  sY    JqMMq1uqd((v  HQ!VTVTV,,!tr1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zgenpareto_gen._pdfh
  r  r1   c                 D    t          ||k    |dk    z  ||fd |           S )Nr   c                 @    t          j        |dz   || z             |z  S r  r  rd   r  s     r/   r  z'genpareto_gen._logpdf.<locals>.<lambda>n
  s"    
1r61Q3(?(?'?!'C r1   r   r  s      r/   r   zgenpareto_gen._logpdfl
  s4    16a1f-1vCC"  	r1   c                 2    t          j        | |            S rA   )rj   inv_boxcox1pr  s      r/   rh   zgenpareto_gen._cdfq
  s    QB''''r1   c                 0    t          j        | |           S rA   )rj   
inv_boxcoxr  s      r/   rl   zgenpareto_gen._sft
  s    }aR!$$$r1   c                 D    t          ||k    |dk    z  ||fd |           S )Nr   c                 8    t          j        || z             |z  S rA   r  r  s     r/   r  z&genpareto_gen._logsf.<locals>.<lambda>y
  s    1~'9 r1   r  r  s      r/   r   zgenpareto_gen._logsfw
  s4    16a1f-1v99"  	r1   c                 2    t          j        | |            S rA   )rj   boxcox1pr  s      r/   rq   zgenpareto_gen._ppf|
  s    QB####r1   c                 0    t          j        ||            S rA   )rj   boxcoxr  s      r/   rt   zgenpareto_gen._isf
  s    	!aR    r1   r  c                 V   d|vrd }n"t          |dk     |fd t          j                  }d|vrd }n"t          |dk     |fd t          j                  }d|vrd }n"t          |dk     |fd	 t          j                  }d
|vrd }n"t          |dk     |fd t          j                  }||||fS )NrT  r   c                     dd| z
  z  S rQ   rz   xis    r/   r  z&genpareto_gen._stats.<locals>.<lambda>
  s    aRj r1   r  r   c                 *    dd| z
  dz  z  dd| z  z
  z  S r"  rz   r  s    r/   r  z&genpareto_gen._stats.<locals>.<lambda>
  s    a1r6A+oQrT&B r1   r  gUUUUUU?c                 Z    dd| z   z  t          j        dd| z  z
            z  dd| z  z
  z  S )NrH   r   r  r  r  s    r/   r  z&genpareto_gen._stats.<locals>.<lambda>
  s5    a1r6lRWQ2X5F5F&F'(1R4x'1 r1   r  r*  c                 `    ddd| z  z
  z  d| dz  z  | z   dz   z  dd| z  z
  z  dd| z  z
  z  dz
  S )Nr  r   rH   r  rz   r  s    r/   r  z&genpareto_gen._stats.<locals>.<lambda>
  sQ    a1qt8n"a%"q8H&I'(1R4x'145"H'>@A'B r1   r   rC   r\   r  )r:   r  r  rT  r  r  r  s          r/   r   zgenpareto_gen._stats
  s    gAA1q51$006# #A gAA1s7QDBB6# #A gAA1s7QD1 16# #A gAA1s7QDB B6# #A !Qzr1   c           	      l    d t          |dk    |ffdt          j        dz                       S )Nc                    d}t          j        d| dz             }t          |t          j        | |                    D ]\  }}||d|z  z  d||z  z
  z  z   }t          j        || z  dk     |d|z  | z  z  t           j                  S )Nr{   r   r   r  r|   r  )rC   r  ziprj   combr  r\   )rU   r  r  r  kicnks         r/   r	  z#genpareto_gen._munp.<locals>.__munp
  s    C	!QU##Aq"'!Q--00 > >CC2"*,a"f==8AEAIsdQh1_'<bfEEEr1   r   c                      |           S rA   rz   )r  _genpareto_gen__munprU   s    r/   r  z%genpareto_gen._munp.<locals>.<lambda>
  s    FF1aLL r1   r   )r   rj   r  )r:   rU   r  r  s    ` @r/   r   zgenpareto_gen._munp
  sR    	F 	F 	F !q&1$00000(1q5//+ + 	+r1   c                     d|z   S r  rz   r)  s     r/   r   zgenpareto_gen._entropy
  s    Avr1   Nr  )rv   rw   rx   ry   rV   r^   r   re   r   rh   rl   r   rq   rt   r   r   r   rz   r1   r/   r  r  6
  s        " "F  K K K  * * *  
( ( (% % %  
$ $ $! ! !   :	+ 	+ 	+    r1   r  	genparetoc                   0    e Zd ZdZd Zd Zd Zd Zd ZdS )genexpon_gena  A generalized exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genexpon` is:

    .. math::

        f(x, a, b, c) = (a + b (1 - \exp(-c x)))
                        \exp(-a x - b x + \frac{b}{c}  (1-\exp(-c x)))

    for :math:`x \ge 0`, :math:`a, b, c > 0`.

    `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
    Distribution", Journal of the American Statistical Association, 1993.

    N. Balakrishnan, "The Exponential Distribution: Theory, Methods and
    Applications", Asit P. Basu.

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }t          dddt          j        fd          }|||gS )Nr~   Fr   r   r   r  r[   )r:   rZ  r[  r  s       r/   r^   zgenexpon_gen._shape_info
  sY    UQK@@UQK@@UQK@@B|r1   c           	          ||t          j        | |z             z  z   t          j        | |z
  |z  |t          j        | |z             z  |z  z             z  S rA   rj   r  rC   r   r:   rd   r~   r   r  s        r/   re   zgenexpon_gen._pdf
  sl     A!A''!Aq01BHaRTNN?0CA0E1F *G *G G 	Gr1   c                     t          j        ||t          j        | |z             z  z             | |z
  |z  z   |t          j        | |z             z  |z  z   S rA   rC   r   rj   r  r  s        r/   r   zgenexpon_gen._logpdf
  sZ    vaBHaRTNN?++,,1ax7BHaRTNN?8KA8MMMr1   c                 z    t          j        | |z
  |z  |t          j        | |z             z  |z  z              S rA   r  r  s        r/   rh   zgenexpon_gen._cdf
  s>    1"Q$A!A$7$99::::r1   c                 x    t          j        | |z
  |z  |t          j        | |z             z  |z  z             S rA   r  r  s        r/   rl   zgenexpon_gen._sf
  s;    vr!tQhRXqbd^^O!4Q!66777r1   N)	rv   rw   rx   ry   r^   re   r   rh   rl   rz   r1   r/   r  r  
  so         <  G G GN N N; ; ;8 8 8 8 8r1   r  genexponc                   v     e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Z fdZd Zd Z xZS )genextreme_genaB  A generalized extreme value continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r

    Notes
    -----
    For :math:`c=0`, `genextreme` is equal to `gumbel_r` with
    probability density function

    .. math::

        f(x) = \exp(-\exp(-x)) \exp(-x),

    where :math:`-\infty < x < \infty`.

    For :math:`c \ne 0`, the probability density function for `genextreme` is:

    .. math::

        f(x, c) = \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1},

    where :math:`-\infty < x \le 1/c` if :math:`c > 0` and
    :math:`1/c \le x < \infty` if :math:`c < 0`.

    Note that several sources and software packages use the opposite
    convention for the sign of the shape parameter :math:`c`.

    `genextreme` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 *    t          j        |          S rA   r  r)  s     r/   rV   zgenextreme_gen._argcheck  r  r1   c                 V    t          ddt          j         t          j        fd          gS r  r[   r]   s    r/   r^   zgenextreme_gen._shape_info  r  r1   c                 
   t          j        |dk    dt          j        |t                    z  t           j                  }t          j        |dk     dt          j        |t                     z  t           j                   }||fS Nr   r|   )rC   r  maximumr   r\   minimum)r:   r  _b_as       r/   r   zgenextreme_gen._get_support  sc    Xa!eS2:a#7#77@@Xa!eS2:a%#8#8826'BB2vr1   c                 D    t          ||k    |dk    z  ||fd |           S )Nr   c                 8    t          j        | | z            |z  S rA   r  r  s     r/   r  z+genextreme_gen._loglogcdf.<locals>.<lambda>  s    rx1~~a'7 r1   r  r  s      r/   
_loglogcdfzgenextreme_gen._loglogcdf  s3    16a1f-1v77!= = 	=r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zgenextreme_gen._pdf   s"     vdll1a(()))r1   c                    t          ||k    |dk    z  ||fd d          }t          j        |           }|                     ||          }t	          j        |          }t	          j        ||dk    |t          j         k    z  d           t          |dk    |t          j         k    z   |||fd t          j                   }t	          j        ||dk    |dk    z  d           |S )Nr   c                     || z  S rA   rz   r  s     r/   r  z(genextreme_gen._logpdf.<locals>.<lambda>'  s
    !A# r1   r{   r   c                     |  |z   |z
  S rA   rz   )pex2lpex2lex2s      r/   r  z(genextreme_gen._logpdf.<locals>.<lambda>/  s    teemd6J r1   r  )r   rj   r  r!  rC   r   putmaskr\   )r:   rd   r  cxlogex2logpex2r&  logpdfs           r/   r   zgenextreme_gen._logpdf&  s    aAF+aV5E5EsKK2#//!Q''vg

7Q!VbfW5s;;;rQw2"&=9:!7F3JJ')vg/ / / 	
6AFqAv.444r1   c                 T    t          j        |                     ||                     S rA   )rC   r   r!  r  s      r/   r   zgenextreme_gen._logcdf4  s#    tq!,,----r1   c                 R    t          j        |                     ||                    S rA   rC   r   r   r  s      r/   rh   zgenextreme_gen._cdf7  r  r1   c                 T    t          j        |                     ||                     S rA   )rj   r  r   r  s      r/   rl   zgenextreme_gen._sf:  s#    a++,,,,r1   c                     t          j        t          j        |                      }t          ||k    |dk    z  ||fd |          S )Nr   c                 :    t          j        | | z             |z  S rA   r  r  s     r/   r  z%genextreme_gen._ppf.<locals>.<lambda>@      !a(8(8'81'< r1   )rC   r   r   r:   rp   r  rd   s       r/   rq   zgenextreme_gen._ppf=  sP    VRVAYYJ16a1f-1v<<aA A 	Ar1   c                     t          j        t          j        |                       }t	          ||k    |dk    z  ||fd |          S )Nr   c                 :    t          j        | | z             |z  S rA   r  r  s     r/   r  z%genextreme_gen._isf.<locals>.<lambda>E  r4  r1   )rC   r   rj   r  r   r5  s       r/   rt   zgenextreme_gen._isfB  sR    VRXqb\\M"""16a1f-1v<<aA A 	Ar1   c                    fd} |d          } |d          } |d          } |d          }t          j        t                    dk     t           j        z  dz  dz  ||dz  z
            t          j        t                    dk     t           j        dz  dz  t	          j        t	          j        dz  d	z             dt	          j        d	z             z  z
            dz  z            }d
}t          j        t                    |k     t           t	          j        t	          j        dz                       z            }	t          j        dk     t           j        |	           }
t          j        dk     t           j        |dz  |z            }t          dk    |||ffdt           j                  }t          j        t                    |dz  k    dt          j
        d          z  t          z  t           j        dz  z  |          }t          dk    ||||fd t           j                  }t          j        t                    |dz  k    d|dz
            }|
|||fS )Nc                 8    t          j        | z  dz             S rQ   r  )rU   r  s    r/   r  z'genextreme_gen._stats.<locals>.<lambda>H  s    bhqsQw'' r1   r   rH   r  r  gHz>r   r  r|   +=r  rq  gUUUUUUտc                 X    t          j        |           | |dz  z   |z  z   z  dz  z  S NrH   ri  rB   )r  r7  r8  g3g2gm12g2mg12s        r/   r  z'genextreme_gen._stats.<locals>.<lambda>Y  s5    WQZZ"QvXr/A)AB63;N r1   r  g(\?r  r  g      пc                 <    |d|z  d||z   z  | z  z   | z  z   |dz  z  S )Nr  rH   rz   )r7  r8  r=  g4r?  s        r/   r  z'genextreme_gen._stats.<locals>.<lambda>a  s3     BrEArF{OB,>$>#BBFAIM r1   gq=
ףp?333333@r  )rC   r  r  r   rj   r  rW  r   r  r   r   r    )r:   r  gr7  r8  r=  rB  gam2kepsgamkrT  r  sk1r  ku1r  r?  s    `              @r/   r   zgenextreme_gen._statsG  s[   ''''QqTTQqTTQqTTQqTT#a&&4-!BE'C);RCZHHQ$s
3"*SU3Y"7"7"*QW:M:M8M"MNNqRUvUW WxAvgrx
1q58I8I/J/J1/LMMHQXrvu--HQXrvr3wu}55 eRR0O O O O#%6	+ + +
 Xc!ffT	)2bgajj=+?q+H#NN eb"b&1N N#%6	+ + +
 Xc!fft+Xs3w??!R|r1   c                     t          |          }|dk     rd}nd}t                                          ||f          S )Nr   r   rq  r  r   r7   r  )r:   r   rD  r~   r  s       r/   r  zgenextreme_gen._fitstartg  sB    $KKq55AAAww  QD 111r1   c                 &   t          j        d|dz             }d||z  z  t          j        t          j        ||          d|z  z  t          j        ||z  dz             z  d          z  }t          j        ||z  dk    |t           j                  S )Nr   r   r|   r  axis)rC   r  r  rj   r  r  r  r\   )r:   rU   r  r  r  s        r/   r   zgenextreme_gen._munpp  s    Ia11a4x"&GAqMMR!G#bhqsQw&7&77    x!b$///r1   c                 "    t           d|z
  z  dz   S rQ   r   r)  s     r/   r   zgenextreme_gen._entropyw  s    q1u~!!r1   )rv   rw   rx   ry   rV   r^   r   r!  re   r   r   rh   rl   rq   rt   r   r  r   r   r  r  s   @r/   r  r  
  s       % %L  K K K  
= = =
* * *  . . .* * *- - -A A A
A A A
  @2 2 2 2 20 0 0" " " " " " "r1   r  
genextremec                 Z    d} fd} dk    r7t          j                   dz   } dk     rt          j        ||d          }|S n* dk    rt          j         d	z            d
z   }n	d  |z
  z  }t          j        ||dd          \  }}}}|dk    rt          d z            |d         S )af  Inverse of the digamma function (real positive arguments only).

    This function is used in the `fit` method of `gamma_gen`.
    The function uses either optimize.fsolve or optimize.newton
    to solve `sc.digamma(x) - y = 0`.  There is probably room for
    improvement, but currently it works over a wide range of y:

    >>> import numpy as np
    >>> rng = np.random.default_rng()
    >>> y = 64*rng.standard_normal(1000000)
    >>> y.min(), y.max()
    (-311.43592651416662, 351.77388222276869)
    >>> x = [_digammainv(t) for t in y]
    >>> np.abs(sc.digamma(x) - y).max()
    1.1368683772161603e-13

    gox?c                 2    t          j        |           z
  S rA   )rj   digammard   ys    r/   r  z_digammainv.<locals>.<lambda>  s    RZ]]Q& r1   g      r   
   g|=)tolg-@g뭁,?r|   dy=T)xtolr  r   z"_digammainv: fsolve failed, y = %rr   )rC   r   r   newtonr  RuntimeError)rV  _emrO  x0valuer  r  rH  s   `       r/   _digammainvra  ~  s    $ &C&&&&D6zzVAYY_r66 OD"%888EL  
RVAeG__w&QBH%_T2E9=? ? ?E4d
axx?!CDDD8Or1   c                        e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Z fdZ eed           fd            Z xZS )	gamma_gena  A gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    erlang, expon

    Notes
    -----
    The probability density function for `gamma` is:

    .. math::

        f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
    gamma function.

    `gamma` takes ``a`` as a shape parameter for :math:`a`.

    When :math:`a` is an integer, `gamma` reduces to the Erlang
    distribution, and when :math:`a=1` to the exponential distribution.

    Gamma distributions are sometimes parameterized with two variables,
    with a probability density function of:

    .. math::

        f(x, \alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}

    Note that this parameterization is equivalent to the above, with
    ``scale = 1 / beta``.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r   r[   r]   s    r/   r^   zgamma_gen._shape_info  r   r1   Nc                 .    |                     ||          S rA   standard_gamma)r:   r~   r   r   s       r/   r   zgamma_gen._rvs  s    **1d333r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zgamma_gen._pdf  r  r1   c                 b    t          j        |dz
  |          |z
  t          j        |          z
  S r  )rj   rf  rW  r  s      r/   r   zgamma_gen._logpdf  s*    x#q!!A%
155r1   c                 ,    t          j        ||          S rA   r[  r  s      r/   rh   zgamma_gen._cdf  ru   r1   c                 ,    t          j        ||          S rA   r^  r  s      r/   rl   zgamma_gen._sf  s    |Aq!!!r1   c                 ,    t          j        ||          S rA   r  r  s      r/   rq   zgamma_gen._ppf  s    ~a###r1   c                 ,    t          j        ||          S rA   rj   rg  r  s      r/   rt   zgamma_gen._isf  s    q!$$$r1   c                 >    ||dt          j        |          z  d|z  fS )Nr   r  r  r  s     r/   r   zgamma_gen._stats  s!    !S^SU**r1   c                 f    t          j        |          d|z
  z  |z   t          j        |          z   S rQ   rj   rL  rW  r  s     r/   r   zgamma_gen._entropy  s*    vayy!A#"RZ]]22r1   c                 |    ddt          |          dz  z   z  }t                                          ||f          S )Nr  :0yE>rH   r  rK  r:   r   r~   r  s      r/   r  zgamma_gen._fitstart  s=     dQ&'ww  QD 111r1   a<          When the location is fixed by using the argument `floc`
        and `method='MLE'`, this
        function uses explicit formulas or solves a simpler numerical
        problem than the full ML optimization problem.  So in that case,
        the `optimizer`, `loc` and `scale` arguments are ignored.
        

r   c                    |                     dd           }|                     dd          }||                                dk    r t                      j        |g|R i |S |                    dd            t          |g d          }|                    dd           }t          |           ||t          d          t          j	        |          }t          j
        |                                          st          d          t          j        ||k              rt          d	|t          j        
          |dk    r||z
  }|                                }|||}	nt          j        |          t          j        |                                          z
  fd}
dz
  t          j        dz
  dz  dz  z             z   dz  z  }|dz  }|dz  }t%          j        |
||d          }	||	z  }nLt          j        |                                          t          j        |          z
  }t)          |          }	|}|	||fS )Nr   r*   r4   r  r  r   r   r   r  r  r   c                 \    t          j        |           t          j        |           z
  z
  S rA   )rC   r   rj   rT  )r~   r  s    r/   r  zgamma_gen.fit.<locals>.<lambda><  s!    RZ]]!:Q!> r1   r  rH   r  r  g333333?gffffff?)disp)r5   r6   r7   r9   r,   r   r0   r   rC   r   r   r   r  r<  r\   r   r   r   r   brentqra  )r:   r   r;   r.   r   r*   r  r   r  r~   rO  aestxar  r(   r  r  r  s                   @r/   r9   zgamma_gen.fit  s^    xx%%(E**<6<<>>T11577;t3d333d333 	!$(=(=(=>>(D))$T*** >f0  ) * * * z${4  $$&& 	ECDDD6$$, 	Bwd"&AAAA199 $;Dyy{{ >~ F4LL26$<<#4#4#6#66>>>>!bgqsQhAo6662a4@5\5\OD"bq999 1HEE
 t!!##bfVnn4AAAE$~r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   rq   rt   r   r   r  r   r   r9   r  r  s   @r/   rc  rc    s+       & &NE E E4 4 4 4* * *6 6 6! ! !" " "$ $ $% % %+ + +3 3 32 2 2 2 2 } 5   I I I I I I I I Ir1   rc  r  c                   ^     e Zd ZdZd Zd Z fdZ eed           fd            Z	 xZ
S )
erlang_gena  An Erlang continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma

    Notes
    -----
    The Erlang distribution is a special case of the Gamma distribution, with
    the shape parameter `a` an integer.  Note that this restriction is not
    enforced by `erlang`. It will, however, generate a warning the first time
    a non-integer value is used for the shape parameter.

    Refer to `gamma` for examples.

    c                     t          j        t          j        |          |k              }|st          j        d|dt
                     |dk    S )NzRThe shape parameter of the erlang distribution has been given a non-integer value .r   )rC   r   floorrp  warnRuntimeWarning)r:   r~   allints      r/   rV   zerlang_gen._argcheckg  sY    q()) 	  MM<=AA@      1ur1   c                 @    t          dddt          j        fd          gS )Nr~   Tr   rZ   r[   r]   s    r/   r^   zerlang_gen._shape_infor  r_   r1   c                     t          ddt          |          dz  z   z            }t          t          |                               ||f          S )Nr  rs  rH   r  )r  r   r7   rc  r  rt  s      r/   r  zerlang_gen._fitstartu  sK     teDkk1n,-..Y%%//A4/@@@r1   a          The Erlang distribution is generally defined to have integer values
        for the shape parameter.  This is not enforced by the `erlang` class.
        When fitting the distribution, it will generally return a non-integer
        value for the shape parameter.  By using the keyword argument
        `f0=<integer>`, the fit method can be constrained to fit the data to
        a specific integer shape parameter.r   c                 >     t                      j        |g|R i |S rA   )r7   r9   r:   r   r;   r.   r  s       r/   r9   zerlang_gen.fit~  s+     uww{4/$///$///r1   )rv   rw   rx   ry   rV   r^   r  r   r   r9   r  r  s   @r/   r|  r|  S  s         &	 	 	C C CA A A A A } 5/ 0 0 00 0 0 00 00 0 0 0 0r1   r|  erlangc                   V    e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
d
 Zd Zd Zd ZdS )gengamma_gena  A generalized gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma, invgamma, weibull_min

    Notes
    -----
    The probability density function for `gengamma` is ([1]_):

    .. math::

        f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gengamma` takes :math:`a` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] E.W. Stacy, "A Generalization of the Gamma Distribution",
       Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192.

    %(example)s

    c                     |dk    |dk    z  S r:  rz   )r:   r~   r  s      r/   rV   zgengamma_gen._argcheck      A!q&!!r1   c                     t          dddt          j        fd          }t          ddt          j         t          j        fd          }||gS r  r[   r  s      r/   r^   zgengamma_gen._shape_info  B    UQK@@UbfWbf$5~FFBxr1   c                 T    t          j        |                     |||                    S rA   r  r  s       r/   re   zgengamma_gen._pdf  "    vdll1a++,,,r1   c                 `    t          |dk    |dk    z  ||ffdt          j                   S )Nr   c                     t          j        t          |                    t          j        |z  dz
  |           z   | |z  z
  t          j                  z
  S rQ   )rC   r   r  rj   rf  rW  )rd   r  r~   s     r/   r  z&gengamma_gen._logpdf.<locals>.<lambda>  sJ    s1vv!A#'19M9M(M*+Q$)/13A)? r1   r  r   rC   r\   r  s     ` r/   r   zgengamma_gen._logpdf  sN    16a!e,q!f@ @ @ @%'VG- - - 	-r1   c                     ||z  }t          j        ||          }t          j        ||          }t          j        |dk    ||          S r:  rj   r\  r_  rC   r  r:   rd   r~   r  xcval1val2s          r/   rh   zgengamma_gen._cdf  E    T{1b!!|Ar""xAtT***r1   Nc                 @    |                     ||          }|d|z  z  S )Nr  r|   rf  )r:   r~   r  r   r   rs         r/   r   zgengamma_gen._rvs  s(    '''552a4yr1   c                     ||z  }t          j        ||          }t          j        ||          }t          j        |dk    ||          S r:  r  r  s          r/   rl   zgengamma_gen._sf  r  r1   c                     t          j        ||          }t          j        ||          }t          j        |dk    ||          d|z  z  S r  rj   rc  rg  rC   r  r:   rp   r~   r  r  r  s         r/   rq   zgengamma_gen._ppf  E    ~a##q!$$xAtT**SU33r1   c                     t          j        ||          }t          j        ||          }t          j        |dk    ||          d|z  z  S r  r  r  s         r/   rt   zgengamma_gen._isf  r  r1   c                 8    t          j        ||dz  |z            S r  )rj   poch)r:   rU   r~   r  s       r/   r   zgengamma_gen._munp  s    wq!C%'"""r1   c                     t          j        |          }|d|z
  z  d|z  |z  z   t          j        |          z   t          j        t          |                    z
  S r  )rj   rL  rW  rC   r   r  )r:   r~   r  r  s       r/   r   zgengamma_gen._entropy  sJ    fQii!C%y3q59$rz!}}4rvc!ff~~EEr1   r   )rv   rw   rx   ry   rV   r^   re   r   rh   r   rl   rq   rt   r   r   rz   r1   r/   r  r    s         >" " "  
- - -- - -+ + +   + + +4 4 4
4 4 4
# # #F F F F Fr1   r  gengammac                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	genhalflogistic_gena  A generalized half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genhalflogistic` is:

    .. math::

        f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}

    for :math:`0 \le x \le 1/c`, and :math:`c > 0`.

    `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zgenhalflogistic_gen._shape_info  r   r1   c                     | j         d|z  fS r  r~   r)  s     r/   r   z genhalflogistic_gen._get_support  s    vs1u}r1   c                 v    d|z  }t          j        d||z  z
            }||dz
  z  }||z  }d|z  d|z   dz  z  S rC  rC   r   )r:   rd   r  limitr-  tmp0tmp2s          r/   re   zgenhalflogistic_gen._pdf  sQ     Aj1Q3U1W~Cxv4!##r1   c                 `    d|z  }t          j        d||z  z
            }||z  }d|z
  d|z   z  S r  r  )r:   rd   r  r  r-  r  s         r/   rh   zgenhalflogistic_gen._cdf  s>    Aj1Q3U|DQtV$$r1   c                 0    d|z  dd|z
  d|z   z  |z  z
  z  S r  rz   r  s      r/   rq   zgenhalflogistic_gen._ppf  s'    1ua#a%#a%1,,--r1   c                 B    dd|z  dz   t          j        d          z  z
  S NrH   r   r#  r)  s     r/   r   zgenhalflogistic_gen._entropy  s"    AaCE26!99$$$r1   N)
rv   rw   rx   ry   r^   r   re   rh   rq   r   rz   r1   r/   r  r    s{         *E E E  $ $ $% % %. . .% % % % %r1   r  genhalflogisticc                   N     e Zd ZdZd Zd Z fdZd Zd Zd Z	dd	Z
d
 Z xZS )genhyperbolic_genu  A generalized hyperbolic continuous random variable.

    %(before_notes)s

    See Also
    --------
    t, norminvgauss, geninvgauss, laplace, cauchy

    Notes
    -----
    The probability density function for `genhyperbolic` is:

    .. math::

        f(x, p, a, b) =
            \frac{(a^2 - b^2)^{p/2}}
            {\sqrt{2\pi}a^{p-0.5}
            K_p\Big(\sqrt{a^2 - b^2}\Big)}
            e^{bx} \times \frac{K_{p - 1/2}
            (a \sqrt{1 + x^2})}
            {(\sqrt{1 + x^2})^{1/2 - p}}

    for :math:`x, p \in ( - \infty; \infty)`,
    :math:`|b| < a` if :math:`p \ge 0`,
    :math:`|b| \le a` if :math:`p < 0`.
    :math:`K_{p}(.)` denotes the modified Bessel function of the second
    kind and order :math:`p` (`scipy.special.kn`)

    `genhyperbolic` takes ``p`` as a tail parameter,
    ``a`` as a shape parameter,
    ``b`` as a skewness parameter.

    %(after_notes)s

    The original parameterization of the Generalized Hyperbolic Distribution
    is found in [1]_ as follows

    .. math::

        f(x, \lambda, \alpha, \beta, \delta, \mu) =
           \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}
           e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}
           (\alpha \sqrt{\delta^2 + (x - \mu)^2})}
           {(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}}

    for :math:`x \in ( - \infty; \infty)`,
    :math:`\gamma := \sqrt{\alpha^2 - \beta^2}`,
    :math:`\lambda, \mu \in ( - \infty; \infty)`,
    :math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`,
    :math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`.

    The location-scale-based parameterization implemented in
    SciPy is based on [2]_, where :math:`a = \alpha\delta`,
    :math:`b = \beta\delta`, :math:`p = \lambda`,
    :math:`scale=\delta` and :math:`loc=\mu`

    Moments are implemented based on [3]_ and [4]_.

    For the distributions that are a special case such as Student's t,
    it is not recommended to rely on the implementation of genhyperbolic.
    To avoid potential numerical problems and for performance reasons,
    the methods of the specific distributions should be used.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions
       on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
       pp. 151-157, 1978. https://www.jstor.org/stable/4615705

    .. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model:
        Financial Derivatives and Risk Measures. In: Geman H., Madan D.,
        Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier
        Congress 2000. Springer Finance. Springer, Berlin, Heidelberg.
        :doi:`10.1007/978-3-662-12429-1_12`

    .. [3] Scott, David J, Würtz, Diethelm, Dong, Christine and Tran,
       Thanh Tam, (2009), Moments of the generalized hyperbolic
       distribution, MPRA Paper, University Library of Munich, Germany,
       https://EconPapers.repec.org/RePEc:pra:mprapa:19081.

    .. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic
       and inverse Gaussian distributions: Limiting cases and approximation
       of processes. FDM Preprint 80, April 2003. University of Freiburg.
       https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content

    %(example)s

    c                     t          j        t          j        |          |k     |dk              t          j        t          j        |          |k    |dk               z  S r:  )rC   logical_andr  )r:   r}  r~   r   s       r/   rV   zgenhyperbolic_gen._argcheckr  sJ    rvayy1}a1f55.aQ778 	9r1   c                     t          ddt          j         t          j        fd          }t          dddt          j        fd          }t          ddt          j         t          j        fd          }|||gS )Nr}  Fr   r~   r   rZ   r   r[   )r:   iprZ  r[  s       r/   r^   zgenhyperbolic_gen._shape_infov  sc    UbfWbf$5~FFUQK??UbfWbf$5~FFB|r1   c                 J    t                                          |d          S )N)r   r   r   r  r  r  s     r/   r  zgenhyperbolic_gen._fitstart|  s     ww  K 888r1   c                 H    t           j        d             } |||||          S )Nc                 0    t          j        | |||          S rA   )r   genhyperbolic_logpdfrd   r}  r~   r   s       r/   _logpdf_singlez1genhyperbolic_gen._logpdf.<locals>._logpdf_single  s    .q!Q:::r1   rC   	vectorize)r:   rd   r}  r~   r   r  s         r/   r   zgenhyperbolic_gen._logpdf  s7     
	; 	; 
	; ~aAq)))r1   c                 H    t           j        d             } |||||          S )Nc                 0    t          j        | |||          S rA   )r   genhyperbolic_pdfr  s       r/   _pdf_singlez+genhyperbolic_gen._pdf.<locals>._pdf_single  s    +Aq!Q777r1   r  )r:   rd   r}  r~   r   r  s         r/   re   zgenhyperbolic_gen._pdf  s7     
	8 	8 
	8 {1aA&&&r1   c                 H    t           j        d             } |||||          S )Nc                 h   t          j        |||gt                    j                            t          j                  }t          j        t          d|          }t          j
        |t           j         |           d         }t          j        |          rd}t          j        |t                     |S )N_genhyperbolic_pdfr   zdInfinite values encountered in scipy.special.kve. Values replaced by NaN to avoid incorrect results.)rC   arrayr  ctypesdata_asc_void_pr   from_cythonr   r   quadr\   isnanrp  r  r  )rd   r}  r~   r   	user_datallct1rL   s           r/   _cdf_singlez+genhyperbolic_gen._cdf.<locals>._cdf_single  s    Aq	5 11  #.,i C bfWa003Bx|| 3Lc>222Ir1   r  )r:   rd   r}  r~   r   r  s         r/   rh   zgenhyperbolic_gen._cdf  s5    		 	 
	" {1aA&&&r1   Nc                 \   t          j        |d          t          j        |d          z
  }t          j        |d          }t          j        |d          }t                              |||||          }	t                              ||          }
||	z  t          j        |	          |
z  z   S )NrH   r   rq  )r}  r   r(   r   r   r  )rC   float_powergeninvgaussr  r   r   )r:   r}  r~   r   r   r   r  t2t3gignormsts              r/   r   zgenhyperbolic_gen._rvs  s    
 ^Aq!!BN1a$8$88^B$$^B&&oo%    t,??3w...r1   c                 v   t          j        |||          \  }}}t          j        |d          t          j        |d          z
  }t          j        |d          }t          j        dd          t          j        |d          z  }t          j        ddd          }|                    |j        d|j        z  z             }t          j        ||z   |          \  }}}	}
fd	|||	|
fD             \  }}}}||z  |z  }||z  t          j        |d          t          j        |d          z  |t          j        |d          z
  z  z   }t          j        |d
          t          j        |d
          z  |d
|z  |z  t          j        d          z  z
  dt          j        |d
          z  z   z  d
|z  t          j        |d          z  |t          j        |d          z
  z  z   }|t          j        |d          z  }t          j        |d          t          j        |d          z  |d|	z  |z  t          j        d          z  z
  d|z  t          j        |d          z  t          j        d          z  z   d
t          j        |d          z  z
  z  t          j        |d          t          j        |d
          z  d|z  d|z  |z  t          j        d          z  z
  dt          j        |d
          z  z   z  z   d
t          j        |d          z  |z  z   }|t          j        |d          z  d
z
  }||||fS )NrH   r   r   r  r   r  r  )r   c                     g | ]}|z  S rz   rz   ).0r   b0s     r/   
<listcomp>z,genhyperbolic_gen._stats.<locals>.<listcomp>  s    ;;;Q!b&;;;r1   r  r  r3  r  rY  r  )	rC   broadcast_arraysr  linspacer  shaper  rj   kv)r:   r}  r~   r   r  r  integersb1b2b3b4r1r2r3r4rT  r  m3er  m4er  r  s                        @r/   r   zgenhyperbolic_gen._stats  sE    %aA..1a^Aq!!BN1a$8$88^B$$^Aq!!BN2s$;$;;;q!Q''##HNTAF]$BCCU1x<44BB;;;;2r2r*:;;;BBFRKGbnQ**R^B-B-BB".Q''') ) 	

 N1a  2>"a#8#88!b&2+r2 6 666A&&&'( EBN2q)))".Q''')) 	 ".G,,,N1a  2>"a#8#88!b&2+r3 7 777VbnR+++bnR.E.EEFA&&&'( N1a  2>"a#8#88Vb2glR^B%<%<<<A&&&'(	( r1%%%*+ 	 ".B'''!+!Qzr1   r   )rv   rw   rx   ry   rV   r^   r  r   re   rh   r   r   r  r  s   @r/   r  r    s        W Wr9 9 9  9 9 9 9 9* * *' ' '' ' ',/ / / /*' ' ' ' ' ' 'r1   r  genhyperbolicc                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	gompertz_genaq  A Gompertz (or truncated Gumbel) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gompertz` is:

    .. math::

        f(x, c) = c \exp(x) \exp(-c (e^x-1))

    for :math:`x \ge 0`, :math:`c > 0`.

    `gompertz` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zgompertz_gen._shape_info   r   r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zgompertz_gen._pdf  r  r1   c                 `    t          j        |          |z   |t          j        |          z  z
  S rA   r  r  s      r/   r   zgompertz_gen._logpdf  s%    vayy1}q28A;;..r1   c                 X    t          j        | t          j        |          z             S rA   r  r  s      r/   rh   zgompertz_gen._cdf
  s$    !bhqkk)****r1   c                 \    t          j        d|z  t          j        |           z            S r  r  r  s      r/   rq   zgompertz_gen._ppf  s%    xq28QB<</000r1   c                     dt          j        |          z
  t          j        |          t          j        d|          z  z
  S r  )rC   r   r   rj   expnr)  s     r/   r   zgompertz_gen._entropy  s0    RVAYY271a==!888r1   N)
rv   rw   rx   ry   r^   re   r   rh   rq   r   rz   r1   r/   r  r    s{         *E E E* * */ / /+ + +1 1 19 9 9 9 9r1   r  gompertzc                     t          j        |           } t          j        |          }|                                }t          j        ||z
            }t          j        | |          S )N)weights)rC   r   maxr   average)rd   
logweightsmaxlogwr  s       r/   _average_with_log_weightsr    sV    

1AJ''JnnGfZ')**G:a))))r1   c                       e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Ze ee          d                         ZdS )gumbel_r_gena  A right-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_l, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_r` is:

    .. math::

        f(x) = \exp(-(x + e^{-x}))

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zgumbel_r_gen._shape_info9  r   r1   c                 P    t          j        |                     |                    S rA   r  r   s     r/   re   zgumbel_r_gen._pdf<      vdll1oo&&&r1   c                 4    | t          j        |           z
  S rA   r  r   s     r/   r   zgumbel_r_gen._logpdf@  s    rBFA2JJr1   c                 R    t          j        t          j        |                      S rA   r  r   s     r/   rh   zgumbel_r_gen._cdfC  s    vrvqbzzk"""r1   c                 .    t          j        |            S rA   r  r   s     r/   r   zgumbel_r_gen._logcdfF  s    r

{r1   c                 R    t          j        t          j        |                      S rA   r#  r   s     r/   rq   zgumbel_r_gen._ppfI  s    q		z""""r1   c                 T    t          j        t          j        |                       S rA   r  r   s     r/   rl   zgumbel_r_gen._sfL  s!    "&!**%%%%r1   c                 T    t          j        t          j        |                       S rA   rC   r   r  r  s     r/   rt   zgumbel_r_gen._isfO  s!    !}%%%%r1   c                     t           t          j        t          j        z  dz  dt          j        d          z  t          j        dz  z  t          z  dfS )Nr  r  r  r  rC  r   rC   r   r   r    r]   s    r/   r   zgumbel_r_gen._statsR  s9    ruRU{3271::beQh(>(GOOr1   c                     t           dz   S r  rP  r]   s    r/   r   zgumbel_r_gen._entropyU  s    {r1   c                    t          | ||          \  }}fd}||} ||          n|	|fdnfd|                    dd          }|dz  |dz  }
}	fd} ||	|
          sB|	dk    s|
t          j        k     r,|	dz  }	|
dz  }
 ||	|
          s|	dk    |
t          j        k     ,t	          j        |	|
fd	d	
          }|j        }||n
 ||          |fS )Nc                     |  t          j         | z            t          j        t	                              z
  z  S rA   )rj   	logsumexprC   r   r  )r(   r   s    r/   get_loc_from_scalez,gumbel_r_gen.fit.<locals>.get_loc_from_scalef  s5    6R\4%%-8826#d));L;LLMMr1   c                     z
  t          j        z
  | z            z  z   }t                    | z   z  }|                                |z
  S rA   )rC   r   r  r  )r(   term1term2r   r'   s      r/   rO  zgumbel_r_gen.fit.<locals>.funcy  sP     4Z263:2F+G+GG$NEIIu5E 99;;..r1   c                 f     | z  }t          |          }                                |z
  | z
  S )N)r  )r  r   )r(   sdatawavgr   s      r/   rO  zgumbel_r_gen.fit.<locals>.func  s8    !EEME4TeLLLD99;;-55r1   r(   r   rH   c                 |    t          j         |                     t          j         |                    k    S rA   rB   )rE   rF   rO  s     r/   rG   z0gumbel_r_gen.fit.<locals>.interval_contains_root  s7    V--V--. /r1   r   r:  )r  rtolr[  )r  r5   rC   r\   r   r$   r  )r:   r   r;   r.   r   r   r  r(   brack_startrE   rF   rG   resrO  r'   s    `           @@r/   r9   zgumbel_r_gen.fitY  s    9t9=tE EdF	N 	N 	N 	N 	N  E$$U++CC / / / / / / /6 6 6 6 6 ((7A..K(1_kAoFF
/ / / / / .-ff== 

frvoo!! .-ff== 

frvoo &tff5E,1? ? ?CHE*$$0B0B50I0ICEzr1   N)rv   rw   rx   ry   r^   re   r   rh   r   rq   rl   rt   r   r   r>   r
   r   r9   rz   r1   r/   r  r    s         2  ' ' '  # # #  # # #& & && & &P P P   M**@ @ +* _@ @ @r1   r  gumbel_rc                       e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Ze ee          d                         ZdS )gumbel_l_gena  A left-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_l` is:

    .. math::

        f(x) = \exp(x - e^x)

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zgumbel_l_gen._shape_info  r   r1   c                 P    t          j        |                     |                    S rA   r  r   s     r/   re   zgumbel_l_gen._pdf  r  r1   c                 0    |t          j        |          z
  S rA   r  r   s     r/   r   zgumbel_l_gen._logpdf  r+  r1   c                 R    t          j        t          j        |                      S rA   r  r   s     r/   rh   zgumbel_l_gen._cdf  s    "&))$$$$r1   c                 R    t          j        t          j        |                      S rA   rC   r   rj   r  r   s     r/   rq   zgumbel_l_gen._ppf  s    vrx||m$$$r1   c                 ,    t          j        |           S rA   r  r   s     r/   r   zgumbel_l_gen._logsf  r  r1   c                 P    t          j        t          j        |                     S rA   r  r   s     r/   rl   zgumbel_l_gen._sf      vrvayyj!!!r1   c                 P    t          j        t          j        |                     S rA   r#  r   s     r/   rt   zgumbel_l_gen._isf  r*  r1   c                     t            t          j        t          j        z  dz  dt          j        d          z  t          j        dz  z  t          z  dfS )Nr  r  r  rC  r  r]   s    r/   r   zgumbel_l_gen._stats  s@    wbeC271::~beQh&/8 	8r1   c                     t           dz   S r  rP  r]   s    r/   r   zgumbel_l_gen._entropy  s    {r1   c                     |                     d          |d          |d<   t          j        t          j        |           g|R i |\  }}| |fS )Nr   )r5   r  r9   rC   r   )r:   r   r;   r.   loc_rscale_rs         r/   r9   zgumbel_l_gen.fit  sa     88F' L=DL",
4(8(8'8H4HHH4HHwvwr1   N)rv   rw   rx   ry   r^   re   r   rh   rq   r   rl   rt   r   r   r>   r
   r   r9   rz   r1   r/   r!  r!    s         4  ' ' '  % % %% % %  " " "" " "8 8 8   M**  +* _  r1   r!  gumbel_lc                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
halfcauchy_gena  A Half-Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfcauchy` is:

    .. math::

        f(x) = \frac{2}{\pi (1 + x^2)}

    for :math:`x \ge 0`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zhalfcauchy_gen._shape_info  r   r1   c                 2    dt           j        z  d||z  z   z  S r  r   r   s     r/   re   zhalfcauchy_gen._pdf  r1  r1   c                 t    t          j        dt           j        z            t          j        ||z            z
  S r0  rC   r   r   rj   r  r   s     r/   r   zhalfcauchy_gen._logpdf  s)    vc"%i  28AaC==00r1   c                 J    dt           j        z  t          j        |          z  S r0  r3  r   s     r/   rh   zhalfcauchy_gen._cdf  s    25y1%%r1   c                 J    t          j        t           j        dz  |z            S r	  r7  r   s     r/   rq   zhalfcauchy_gen._ppf  s    vbeAgai   r1   c                 ^    t           j        t           j        t           j        t           j        fS rA   r
  r]   s    r/   r   zhalfcauchy_gen._stats  r=  r1   c                 D    t          j        dt           j        z            S r	  r   r]   s    r/   r   zhalfcauchy_gen._entropy  r?  r1   N)rv   rw   rx   ry   r^   re   r   rh   rq   r   r   rz   r1   r/   r4  r4    s         &  # # #1 1 1& & &! ! !. . .    r1   r4  
halfcauchyc                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
halflogistic_genaG  A half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halflogistic` is:

    .. math::

        f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
             = \frac{1}{2} \text{sech}(x/2)^2

    for :math:`x \ge 0`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zhalflogistic_gen._shape_info3  r   r1   c                 P    t          j        |                     |                    S rA   r  r   s     r/   re   zhalflogistic_gen._pdf6  s     vdll1oo&&&r1   c                     t          j        d          |z
  dt          j        t          j        |                     z  z
  S r   )rC   r   rj   r  r   r   s     r/   r   zhalflogistic_gen._logpdf;  s2    vayy1}rBHRVQBZZ$8$8888r1   c                 0    t          j        |dz            S r0  )rC   tanhr   s     r/   rh   zhalflogistic_gen._cdf>  s    wqu~~r1   c                 0    dt          j        |          z  S r	  )rC   arctanhr   s     r/   rq   zhalflogistic_gen._ppfA  s    Ar1   c                 d   |dk    rdt          j        d          z  S |dk    rt           j        t           j        z  dz  S |dk    r
dt          z  S |dk    rdt           j        dz  z  dz  S ddt	          d	d|z
            z
  z  t          j        |dz             z  t          j        |d          z  S )
Nr   rH   r  r  r  r     r  r   )rC   r   r   r    r  rj   r  r  rT   s     r/   r   zhalflogistic_gen._munpD  s    66RVAYY;665;s?"66V8O66RUAX:$$!CQqSMM/"28AaC==0A>>r1   c                 0    dt          j        d          z
  S r	  r#  r]   s    r/   r   zhalflogistic_gen._entropyO  r$  r1   N)rv   rw   rx   ry   r^   re   r   rh   rq   r   r   rz   r1   r/   r?  r?    s         (  ' ' '
9 9 9    	? 	? 	?    r1   r?  halflogisticc                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )halfnorm_genaF  A half-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfnorm` is:

    .. math::

        f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)

    for :math:`x >= 0`.

    `halfnorm` is a special case of `chi` with ``df=1``.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zhalfnorm_gen._shape_infol  r   r1   Nc                 H    t          |                    |                    S Nr  rm  r   s      r/   r   zhalfnorm_gen._rvso  s!    <//T/::;;;r1   c                 |    t          j        dt           j        z            t          j        | |z  dz            z  S r0  rC   r   r   r   r   s     r/   re   zhalfnorm_gen._pdfr  s1    ws25y!!"&!Ac"2"222r1   c                 \    dt          j        dt           j        z            z  ||z  dz  z
  S Nr   r   r   r   s     r/   r   zhalfnorm_gen._logpdfv  s*    RVCI&&&1S00r1   c                 ,    t          |          dz  dz
  S NrH   r|   r   r   s     r/   rh   zhalfnorm_gen._cdfy  s    ||A~c!!r1   c                 6    t          j        d|z   dz            S r  r   r   s     r/   rq   zhalfnorm_gen._ppf|  s    x1c	"""r1   c                    t          j        dt           j        z            ddt           j        z  z
  t          j        d          dt           j        z
  z  t           j        dz
  dz  z  dt           j        dz
  z  t           j        dz
  dz  z  fS )Nr   r   rH   r  ri  r  r  rC   r   r   r]   s    r/   r   zhalfnorm_gen._stats  sm    BE	""#be)

AbeG$beAg^3257RU1WqL(* 	*r1   c                 P    dt          j        t           j        dz            z  dz   S rS  r   r]   s    r/   r   zhalfnorm_gen._entropy  s"    26"%)$$$S((r1   r   rv   rw   rx   ry   r^   r   re   r   rh   rq   r   r   rz   r1   r/   rL  rL  V  s         *  < < < <3 3 31 1 1" " "# # #* * *) ) ) ) )r1   rL  halfnormc                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	hypsecant_gena  A hyperbolic secant continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `hypsecant` is:

    .. math::

        f(x) = \frac{1}{\pi} \text{sech}(x)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zhypsecant_gen._shape_info  r   r1   c                 J    dt           j        t          j        |          z  z  S r  )rC   r   coshr   s     r/   re   zhypsecant_gen._pdf  s    BE"'!**$%%r1   c                 n    dt           j        z  t          j        t          j        |                    z  S r0  )rC   r   r4  r   r   s     r/   rh   zhypsecant_gen._cdf  s%    25y26!99----r1   c                 n    t          j        t          j        t           j        |z  dz                      S r0  )rC   r   r8  r   r   s     r/   rq   zhypsecant_gen._ppf  s&    vbfRU1WS[))***r1   c                 B    dt           j        t           j        z  dz  ddfS )Nr   r  rH   r   r]   s    r/   r   zhypsecant_gen._stats  s    "%+a-A%%r1   c                 D    t          j        dt           j        z            S r	  r   r]   s    r/   r   zhypsecant_gen._entropy  r?  r1   Nr%  rz   r1   r/   r]  r]    sx         &  & & &. . .+ + +& & &    r1   r]  	hypsecantc                   *    e Zd ZdZd Zd Zd Zd ZdS )gausshyper_gena_  A Gauss hypergeometric continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gausshyper` is:

    .. math::

        f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}

    for :math:`0 \le x \le 1`, :math:`a,b > 0`, :math:`c` a real number,
    :math:`z > -1`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
    :math:`F[2, 1]` is the Gauss hypergeometric function
    `scipy.special.hyp2f1`.

    `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
    parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in
           Queues." *Journal of the Royal Statistical Society*. Series D (The
           Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939

    %(example)s

    c                 8    |dk    |dk    z  ||k    z  |dk    z  S )Nr   r  rz   )r:   r~   r   r  r"  s        r/   rV   zgausshyper_gen._argcheck  s'    A!a% AF+q2v66r1   c                    t          dddt          j        fd          }t          dddt          j        fd          }t          ddt          j         t          j        fd          }t          dddt          j        fd          }||||gS )	Nr~   Fr   r   r   r  r"  r  r[   )r:   rZ  r[  r  izs        r/   r^   zgausshyper_gen._shape_info  sz    UQK@@UQK@@UbfWbf$5~FFURL.AABBr1   c                    t          j        |          t          j        |          z  t          j        ||z             z  t          j        ||||z   |           z  }d|z  ||dz
  z  z  d|z
  |dz
  z  z  d||z  z   |z  z  S r  )rj   r  hyp2f1)r:   rd   r~   r   r  r"  Cinvs          r/   re   zgausshyper_gen._pdf  s     x{{28A;;&rx!}}4RYq!QqS1"5M5MM4x!ae*$A3'773qs7Q,FFr1   c                     t          j        ||z   |          t          j        ||          z  }t          j        |||z   ||z   |z   |           }t          j        ||||z   |           }||z  |z  S rA   )rj   r_  rl  )	r:   rU   r~   r   r  r"  r  r  r5  s	            r/   r   zgausshyper_gen._munp  sp    gac1oo1-i1Q3!Ar**i1acA2&&3w}r1   N)rv   rw   rx   ry   rV   r^   re   r   rz   r1   r/   rg  rg    s^         @7 7 7     G G G    r1   rg  
gausshyperc                   X    e Zd ZdZej        Zd Zd Zd Z	d Z
d Zd Zd Zdd
Zd ZdS )invgamma_gena_  An inverted gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgamma` is:

    .. math::

        f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})

    for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `invgamma` takes ``a`` as a shape parameter for :math:`a`.

    `invgamma` is a special case of `gengamma` with ``c=-1``, and it is a
    different parameterization of the scaled inverse chi-squared distribution.
    Specifically, if the scaled inverse chi-squared distribution is
    parameterized with degrees of freedom :math:`\nu` and scaling parameter
    :math:`\tau^2`, then it can be modeled using `invgamma` with
    ``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zinvgamma_gen._shape_info  r   r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zinvgamma_gen._pdf  r  r1   c                 n    |dz    t          j        |          z  t          j        |          z
  d|z  z
  S r  rC   r   rj   rW  r  s      r/   r   zinvgamma_gen._logpdf  s1    1vq		!BJqMM1CE99r1   c                 2    t          j        |d|z            S r  r^  r  s      r/   rh   zinvgamma_gen._cdf  s    |AsQw'''r1   c                 2    dt          j        ||          z  S r  rn  r  s      r/   rq   zinvgamma_gen._ppf   s    R_Q****r1   c                 2    t          j        |d|z            S r  r[  r  s      r/   rl   zinvgamma_gen._sf#  s    {1cAg&&&r1   c                 2    dt          j        ||          z  S r  r  r  s      r/   rt   zinvgamma_gen._isf&  s    R^Aq))))r1   mvskc                 8   t          |dk    |fd t          j                  }t          |dk    |fd t          j                  }d\  }}d|v r"t          |dk    |fd t          j                  }d	|v r"t          |d
k    |fd t          j                  }||||fS )Nr   c                     d| dz
  z  S r  rz   r   s    r/   r  z%invgamma_gen._stats.<locals>.<lambda>*  s    rQV} r1   rH   c                 $    d| dz
  dz  z  | dz
  z  S )Nr|   rH   r   rz   r   s    r/   r  z%invgamma_gen._stats.<locals>.<lambda>+  s    rQVaK/?1r6/J r1   r   r  r  c                 B    dt          j        | dz
            z  | dz
  z  S )Nr  r   r  r  r   s    r/   r  z%invgamma_gen._stats.<locals>.<lambda>2  s     "rwq2v.!b&9 r1   r  r  c                 0    dd| z  dz
  z  | dz
  z  | dz
  z  S )Nr  r  g      &@r  r  rz   r   s    r/   r  z%invgamma_gen._stats.<locals>.<lambda>6  s%    "Q-R8AFC r1   r  )r:   r~   r  m1m2r7  r8  s          r/   r   zinvgamma_gen._stats)  s    At%<%<bfEEAt%J%J    B'>>At9926C CB '>>AtCCRVM MB 2r2~r1   c                 f    ||dz   t          j        |          z  z
  t          j        |          z   S r  rq  r  s     r/   r   zinvgamma_gen._entropy9  s+    AcERVAYY&&A66r1   Nrz  )rv   rw   rx   ry   r   r  r  r^   re   r   rh   rq   rl   rt   r   r   rz   r1   r/   rq  rq    s         : "4ME E E* * *: : :( ( (+ + +' ' '* * *    7 7 7 7 7r1   rq  invgammac                        e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Z fdZ fdZd Z ee           fd            Z xZS )invgauss_gena  An inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2 \pi x^3}}
                    \exp(-\frac{(x-\mu)^2}{2 x \mu^2})

    for :math:`x >= 0` and :math:`\mu > 0`.

    `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS Nr5  Fr   r   r[   r]   s    r/   r^   zinvgauss_gen._shape_infoY  rP  r1   Nc                 2    |                     |d|          S Nr|   r  waldr:   r5  r   r   s       r/   r   zinvgauss_gen._rvs\  s      St 444r1   c                     dt          j        dt           j        z  |dz  z            z  t          j        dd|z  z  ||z
  |z  dz  z            z  S )Nr|   rH   r  r  rQ  r:   rd   r5  s      r/   re   zinvgauss_gen._pdf_  sO     271RU71c6>***26$!*qtRi!^2K+L+LLLr1   c                     dt          j        dt           j        z            z  dt          j        |          z  z
  ||z
  |z  dz  d|z  z  z
  S )Nrq  rH   ri  r   r  s      r/   r   zinvgauss_gen._logpdfd  sF    BF1RU7OO#c"&))m3"by1nac6JJJr1   c                     dt          j        |          z  }t          |||z  dz
  z            }d|z  t          | ||z  dz   z            z   }|t          j        t          j        ||z
                      z   S r"  )rC   r   r   r  r   r:   rd   r5  r  r~   r   s         r/   r   zinvgauss_gen._logcdfk  ss    "'!**nR1-..F\3$1r6Q,"788828BF1q5MM****r1   c                     dt          j        |          z  }t          |||z  dz
  z            }d|z  t          | ||z   z  |z            z   }|t          j        t          j        ||z
                       z   S r"  )rC   r   r   r   r  r   r  s         r/   r   zinvgauss_gen._logsfq  su    "'!**nB!|,--F\3$!b&/B"677728RVAE]]N++++r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   rl   zinvgauss_gen._sfw  s     vdkk!R(()))r1   c                 R    t          j        |                     ||                    S rA   r0  r  s      r/   rh   zinvgauss_gen._cdfz       vdll1b))***r1   c                    t          j        ddd          5  t          j        ||          \  }}t          j        ||d          }|dk    }t          j        d||         z
  ||         d          ||<   t          j        |          }t                                          ||         ||                   ||<   d d d            n# 1 swxY w Y   |S Nr+  )r-  overinvalidr   r   )	rC   r.  r  ra  _invgauss_ppf_invgauss_isfr  r7   rq   )r:   rd   r5  ppfi_wti_nanr  s         r/   rq   zinvgauss_gen._ppf}      [xJJJ 	; 	;'2..EAr&q"a00Cs7D,QqwY4!DDCIHSMMEah5	::CJ	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 	; 
   B"CCCc                    t          j        ddd          5  t          j        ||          \  }}t          j        ||d          }|dk    }t          j        d||         z
  ||         d          ||<   t          j        |          }t                                          ||         ||                   ||<   d d d            n# 1 swxY w Y   |S r  )	rC   r.  r  ra  r  r  r  r7   rt   )r:   rd   r5  isfr  r  r  s         r/   rt   zinvgauss_gen._isf  r  r  c                 D    ||dz  dt          j        |          z  d|z  fS )Nr  r     r  )r:   r5  s     r/   r   zinvgauss_gen._stats  s%    2s7AbgbkkM2b500r1   c                 N   |                     dd          }t          |           t          k    s|                                dk    r t	                      j        |g|R i |S t          | |||          \  }}}}	 || t	                      j        |g|R i |S t          j        ||z
  dk               rt          ddt          j
                  ||z
  }t          j        |          }|-t          |          t          j        |dz  |dz  z
            z  }||z  }|||fS )Nr*   r4   r  r   invgaussr  r  )r5   r8   wald_genr6   r7   r9   r  rC   r  r<  r\   r   r  r  )
r:   r   r;   r.   r*   fshape_sr   r   fshape_nr  s
            r/   r9   zinvgauss_gen.fit  sC   (E**::!!V\\^^t%;%;577;t3d333d333'B4CG(O (O$hf	 <8/577;t3d333d333VD4K!O$$ 	)z"&AAAA$;Dwt}}H~TbfTRZ(b.-H&I&IJ&(Hv%%r1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   r   r   rl   rh   rq   rt   r   r
   r9   r  r  s   @r/   r  r  @  s+        , "4MF F F5 5 5 5M M M
K K K+ + +, , ,* * *+ + +        1 1 1 M**& & & & +*& & & & &r1   r  r  c                   P    e Zd ZdZd Zd Zd Zd Zd Zd Z	dd	Z
d
 Zd Zd ZdS )geninvgauss_genaW  A Generalized Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `geninvgauss` is:

    .. math::

        f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))

    where `x > 0`, `p` is a real number and `b > 0`([1]_).
    :math:`K_p` is the modified Bessel function of second kind of order `p`
    (`scipy.special.kv`).

    %(after_notes)s

    The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of
    `geninvgauss` with `p = -1/2`, `b = 1 / mu` and `scale = mu`.

    Generating random variates is challenging for this distribution. The
    implementation is based on [2]_.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time
       models for the generalized inverse gaussian distribution",
       Stochastic Processes and their Applications 7, pp. 49--54, 1978.

    .. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
       random variates", Statistics and Computing, 24(4), p. 547--557, 2014.

    %(example)s

    c                     ||k    |dk    z  S r:  rz   r:   r}  r   s      r/   rV   zgeninvgauss_gen._argcheck  s    Q1q5!!r1   c                     t          ddt          j         t          j        fd          }t          dddt          j        fd          }||gS )Nr}  Fr   r   r   r[   )r:   r  r[  s      r/   r^   zgeninvgauss_gen._shape_info  B    UbfWbf$5~FFUQK@@Bxr1   c                     t           j        d             } ||||          }t          j        |                                          rd}t	          j        |t                     |S )Nc                 .    t          j        | ||          S rA   )r   geninvgauss_logpdfrd   r}  r   s      r/   logpdf_singlez.geninvgauss_gen._logpdf.<locals>.logpdf_single  s    ,Q1555r1   zjInfinite values encountered in scipy.special.kve(p, b). Values replaced by NaN to avoid incorrect results.)rC   r  r  r  rp  r  r  )r:   rd   r}  r   r  r"  rL   s          r/   r   zgeninvgauss_gen._logpdf  sm     
	6 	6 
	6 M!Q""8A;;?? 	/HCM#~...r1   c                 T    t          j        |                     |||                    S rA   r  r:   rd   r}  r   s       r/   re   zgeninvgauss_gen._pdf  r  r1   c                 ^     | j         | \  }t          j        fd            } ||g|R  S )Nc                     |\  }}t          j        ||gt                    j                            t          j                  }t          j        t          d|          }t          j
        ||           d         S )N_geninvgauss_pdfr   )rC   r  r  r  r  r  r   r  r   r   r  )rd   r;   r}  r   r  r  r  s         r/   r  z)geninvgauss_gen._cdf.<locals>._cdf_single  si    DAq!Q//6>>vOOI".v7I/8: :C >#r1--a00r1   )r   rC   r  )r:   rd   r;   r  r  r  s        @r/   rh   zgeninvgauss_gen._cdf  sU    ""D)B		1 	1 	1 	1 
	1 {1$t$$$$r1   c                 L    t          |dk    |||fd t          j                   S )Nr   c                 T    |dz
  t          j        |           z  || d| z  z   z  dz  z
  S r"  r#  r  s      r/   r  z.geninvgauss_gen._logquasipdf.<locals>.<lambda>
  s,    1q5"&))*;aQqSk!m*K r1   r  r  s       r/   _logquasipdfzgeninvgauss_gen._logquasipdf  s/    !a%!QKK6'# # 	#r1   Nc                   	
 t          j        |          r.t          j        |          r|                     ||||          }nk|j        dk    rI|j        dk    r>|                     |                                |                                ||          }nt          j        ||          \  }}t          |j        |          \  }	t          t          j	        |                    }t          j
        |          }t          j        ||gdgdgdgg          

j        st          	
fdt          t          |           d          D                       }|                     
d         
d         ||                              |          ||<   
                                 
j        |dk    r|                                }|S )Nr   multi_indexreadonlyflagsop_flagsc              3   `   K   | ](}|         sj         |         nt          d           V  )d S rA   r  slicer  jbcits     r/   	<genexpr>z'geninvgauss_gen._rvs.<locals>.<genexpr>:  R       ; ; ! 79eLR^A..t ; ; ; ; ; ;r1   r   rz   )rC   isscalar_rvs_scalarr   r  r  r   r  r  prodemptynditerfinishedtupleranger  r  iternext)r:   r}  r   r   r   outshp
numsamplesidxr  r  s            @@r/   r   zgeninvgauss_gen._rvs  s    ;q>> .	bk!nn .	""1a|<<CCVq[[QVq[[""16688QVVXXt\JJCC &q!,,DAq #17D11GC RWS\\**J (4..CAq6"/&0\J<$@B B BB k   ; ; ; ; ;%*CII:q%9%9; ; ; ; ;++BqE2a5*,8: ::A'#,, C k   2::((**C
r1   c           	         3 d}|sd}dk     r d}                                }d}dk    sdk    rd}n4t          ddt          j        dz
            z  dz            k    rd}nd}t	          t          j        |                    }	t          j        |	          }
t          j        |
          }d}|r|rrddz   z  z  |z
  }d|z  dz
  z  z  dz
  }||dz  dz  z
  }d|dz  z  d	z  ||z  dz  z
  |z   }t          j        | t          j        d
|dz  z            z  dz            }t          j        d|z  dz             }|t          j	        |dz  t          j
        dz  z             z  |dz  z
  }| t          j	        |dz            z  |dz  z
  }                     |          3                     |          3z
  }                     |          3z
  }||z
  t          j        d|z            z  }||z
  t          j        d|z            z  }d}3 fd}|}nt          j        d                     |          z            }dz   t          j        dz   dz  dz  z             z   z  }d}|t          j        d                     |          z            z  }d} fd}||k    rt          d          |dk    rt          d          d}||
k     r|
|z
  }||                    |          z  }|                    |          } |||z
  | z  z   } | |z  |z   }!dt          j        |          z   ||!          k    }"t          j        |"          }#|#dk    r|!|"         ||||#z   <   ||#z  }|dk    r0||
z  dk    r'd                    ||
z            }$t%          |$          |dz  }||
k     ېnމdz
  z  }%t          j        |%dz  f          }&t          j                             |                    }'|'|%z  }(|%dz  k     rNt          j                   })dk    r|)dz  z  |%z  z
  z  z  }*n#|)t          j        ddz  z            z  }*nd\  })}*|&dz
  z  }+d|+z  t          j        |& z  dz            z  z  },|(|*z   |,z   }-||
k     r|
|z
  }t          j        |          t          j        |          }!}.|                    |          }|-|                    |          z  } | |(k    }/t          j        |/          | |(|*z   k    z  }0t          j        |/|0z            }1|%| |/         z  |(z  |!|/<   |'|.|/<   dk    r!|%z  | |0         |(z
  z  |)z  z   dz  z  |!|0<   n8t          j        | |0         |(z
  t          j                  z            z  |!|0<   |)|!|0         dz
  z  z  |.|0<   t          j        |& z  dz            | |1         |(z
  |*z
  z  d|+z  z  z
  }2dz  t          j        |2          z  |!|1<   |+t          j        |!|1          z  dz            z  |.|1<   t          j        ||.z                                 |!          k    }"t!          |"          }#|#dk    r|!|"         ||||#z   <   ||#z  }||
k     t          j        ||	          }!|rd|!z  }!|!S )NFr   r   Tr   rH   r  r     irA  c                 8                         |           z
  S rA   r  )rd   r   lmr}  r:   s    r/   r  z-geninvgauss_gen._rvs_scalar.<locals>.<lambda>}  s    D$5$5aA$>$>$C r1   c                 2                         |           S rA   r  )rd   r   r}  r:   s    r/   r  z-geninvgauss_gen._rvs_scalar.<locals>.<lambda>  s    D$5$5aA$>$> r1   zvmin must be smaller than vmax.zumax must be positive.r  iP  z|Not a single random variate could be generated in {} attempts. Sampling does not appear to work for the provided parameters.)r   r   )_moder  rC   r   r  
atleast_1dr  zerosarccosr  r   r  r   r   r  r   r  r@  r]  r  logical_notr  )4r:   r}  r   r  r   
invert_resrT  
ratio_unif
mode_shiftsize1dNrd   	simulateda2a1p1q1phirM  root1root2d1d2vminvmaxumaxlogqpdfr  xplusir  r  r  r  accept
num_acceptrL   r_  xsk1A1k2A2k3A3Ahcond1cond2cond3r"  r  s4   ```                                                @r/   r  zgeninvgauss_gen._rvs_scalarD  s    
 	Jq55AJJJq! 
66QUUJJ#c1rwq1u~~-12222JJ J r}Z0011GFOOHQKK	 p	, $?1q5\A%)Ua!e_q(1,"a%!)^QY^b2gk1A5ibgcBEk&:&: :Q >??gb2gk***RVC!Gbeai$788826AbfS1Woo-Q6 &&q!Q//&&ua33b8&&ua33b8 	RVC"H%5%55	RVC"H%5%55CCCCCCC vc$"3"3Aq!"<"<<==a%27AEA:1+<#=#==q@rvcD,=,=eQ,J,J&JKKK>>>>>>t|| !BCCCqyy !9:::Aa--	M<//Q/777 ((a(00D4K1,,!eaiBF1II+5VF^^
>><?KAiZ!789+INN1??Evac{{  's+++Q' a--, a!eBQU$$B))!Q2233BbBAEzzVQBZZq55AzBE12Q6BBbfQAX...BBBa!eBR"&"q1---1BR"A a--	M!bhqkk3 ((a(00,,!,444Ru--b2g>uu}55!E(]R/E
%q55"$a%1U8b=A*=*B"Ba!e!LCJJ!"RVQuX]bfQii,G%H%H!HCJE
QU 33%FB37Q;''!qx"}r/A*Ba"f*MM!VbfQii/E
E
{Q': ; ;;%&Q--4+<+<S!Q+G+GG [[
>><?KAiZ!789+I7 a--: jF## 	c'C
r1   c                     |dk     r)|t          j        |dz
  dz  |dz  z             dz   |z
  z  S t          j        d|z
  dz  |dz  z             d|z
  z
  |z  S r"  r  r  s      r/   r  zgeninvgauss_gen._mode  sj    q55Q
QT 122Q6:;;GQUQJA-..!a%8A==r1   c                    t          j        ||z   |          }t          j        ||          }t          j        |          t          j        |          z  }|                                r_d}t          j        |t                     t          j        |t          j	        t          j
                  }||          ||          z  || <   n||z  }|S )NzInfinite values encountered in the moment calculation involving scipy.special.kve. Values replaced by NaN to avoid incorrect results.dtype)rj   kverC   rI   r  rp  r  r  	full_liker  double)	r:   rU   r}  r   r  denominf_valsrL   rT  s	            r/   r   zgeninvgauss_gen._munp  s    fQUAq!8C==28E??2<<>> 	.C M#~...S"&	:::Ay>E8),<<AxiLLeAr1   r   )rv   rw   rx   ry   rV   r^   r   re   rh   r  r   r  r  r   rz   r1   r/   r  r    s        # #H" " "  
  - - -% % %# # #5 5 5 5nS S Sj> > >    r1   r  r  c                   \     e Zd ZdZej        Zd Zd Z fdZ	d Z
d Zd Zdd	Zd
 Z xZS )norminvgauss_gena  A Normal Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norminvgauss` is:

    .. math::

        f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
                     \exp(\sqrt{a^2 - b^2} + b x)

    where :math:`x` is a real number, the parameter :math:`a` is the tail
    heaviness and :math:`b` is the asymmetry parameter satisfying
    :math:`a > 0` and :math:`|b| <= a`.
    :math:`K_1` is the modified Bessel function of second kind
    (`scipy.special.k1`).

    %(after_notes)s

    A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
    can be expressed as a normal mean-variance mixture:
    `Y = b * V + sqrt(V) * X` where `X` is `norm(0,1)` and `V` is
    `invgauss(mu=1/sqrt(a**2 - b**2))`. This representation is used
    to generate random variates.

    Another common parametrization of the distribution (see Equation 2.1 in
    [2]_) is given by the following expression of the pdf:

    .. math::

        g(x, \alpha, \beta, \delta, \mu) =
        \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}
        {\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
        e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}

    In SciPy, this corresponds to
    `a = alpha * delta, b = beta * delta, loc = mu, scale=delta`.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
           Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
           pp. 151-157, 1978.

    .. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and
           Stochastic Volatility Modelling", Scandinavian Journal of
           Statistics, Vol. 24, pp. 1-13, 1997.

    %(example)s

    c                 @    |dk    t          j        |          |k     z  S r:  )rC   absoluter|  s      r/   rV   znorminvgauss_gen._argcheck+  s    A"+a..1,--r1   c                     t          dddt          j        fd          }t          ddt          j         t          j        fd          }||gS rX  r[   rY  s      r/   r^   znorminvgauss_gen._shape_info.  r  r1   c                 J    t                                          |d          S )N)r   r   r  r  r  s     r/   r  znorminvgauss_gen._fitstart3       ww  H 555r1   c                 $   t          j        |dz  |dz  z
            }|t           j        z  t          j        |          z  }t          j        d|          }|t          j        ||z            z  t          j        ||z  ||z  z
            z  |z  S r  )rC   r   r   r   hypotrj   k1e)r:   rd   r~   r   r  fac1sqs          r/   re   znorminvgauss_gen._pdf7  s{    1q!t$$25y26%==(Xa^^bfQVnn$rvacAbDj'9'99B>>r1   c           
      h   t          j        |          r/t          j        | j        |t           j        ||f          d         S g }t          |||          D ]H\  }}}|                    t          j        | j        |t           j        ||f          d                    It          j        |          S )Nr  r   )	rC   r  r   r  re   r\   r  appendr  )r:   rd   r~   r   resultr_  a0r  s           r/   rl   znorminvgauss_gen._sf=  s    ;q>> 	$>$)QaVDDDQGGF #Aq! @ @RinTYBF35r(< < <<=? @ @ @ @8F###r1   c                       fd}t          j        |          r ||||          S g }t          |||          D ]&\  }}}|                     ||||                     't          j        |          S )Nc                    
fd}
                     ||          } |||||           }|dk    r|S |dk    r8d}|}||z   } |||||           dk    rd|z  }||z   } |||||           dk    n7d}|}||z
  } |||||           dk     rd|z  }||z
  } |||||           dk     t          j        |||||| f
j                  }	|	S )Nc                 8                         | ||          |z
  S rA   rl   )rd   r~   r   rp   r:   s       r/   eqz6norminvgauss_gen._isf.<locals>._isf_scalar.<locals>.eqK  s    xx1a((1,,r1   r   r   rH   )r;   r[  )r   r   rx  r[  )rp   r~   r   r%  xmemdeltaleftrightr  r:   s             r/   _isf_scalarz*norminvgauss_gen._isf.<locals>._isf_scalarI  sF   - - - - - 1aBB1aBQww	AvvU
b1a((1,,eGEJE b1a((1,,
 Ezbq!Q''!++eGE:D bq!Q''!++ _RuAq!9*.)5 5 5FMr1   )rC   r  r  r  r  )	r:   rp   r~   r   r+  r  q0r   r  s	   `        r/   rt   znorminvgauss_gen._isfH  s    	 	 	 	 	B ;q>> 	$;q!Q'''F #Aq! 7 7Rkk"b"5566668F###r1   Nc                     t          j        |dz  |dz  z
            }t                              d|z  ||          }||z  t          j        |          t                              ||          z  z   S )NrH   r   )r5  r   r   r  )rC   r   r  r  r   )r:   r~   r   r   r   r  igs          r/   r   znorminvgauss_gen._rvsr  sy     1q!t$$\\QuW4l\KK2vdhhD<H '/ 'J 'J J J 	Jr1   c                     t          j        |dz  |dz  z
            }||z  }|dz  |dz  z  }d|z  |t          j        |          z  z  }ddd|dz  z  |dz  z  z   z  |z  }||||fS )NrH   r  r  r   r  r  )r:   r~   r   r  r   varianceskewnesskurtosiss           r/   r   znorminvgauss_gen._statsz  s    1q!t$$5ya4%(?7a"'%..01!a!Q$hAo-.6Xx11r1   r   )rv   rw   rx   ry   r   r  r  rV   r^   r  re   rl   rt   r   r   r  r  s   @r/   r  r    s        4 4j "4M. . .  
6 6 6 6 6? ? ?	$ 	$ 	$($ ($ ($TJ J J J2 2 2 2 2 2 2r1   r  norminvgaussc                   b     e Zd ZdZej        Zd Zd Zd Z	d Z
d Zd Zd Zd	 Zd fd	Z xZS )invweibull_genu  An inverted Weibull continuous random variable.

    This distribution is also known as the Fréchet distribution or the
    type II extreme value distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invweibull` is:

    .. math::

        f(x, c) = c x^{-c-1} \exp(-x^{-c})

    for :math:`x > 0`, :math:`c > 0`.

    `invweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
    Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zinvweibull_gen._shape_info  r   r1   c                     t          j        || dz
            }t          j        ||           }t          j        |           }||z  |z  S r  rC   rj  r   )r:   rd   r  xc1xc2s        r/   re   zinvweibull_gen._pdf  sG    hq1"s(##hq1"oofcTll3w}r1   c                 X    t          j        ||           }t          j        |           S rA   r8  )r:   rd   r  r9  s       r/   rh   zinvweibull_gen._cdf  s#    hq1"oovsd||r1   c                 6    t          j        || z              S rA   )rC   r  r  s      r/   rl   zinvweibull_gen._sf  s    !aR%    r1   c                 X    t          j        t          j        |           d|z            S r  )rC   rj  r   r  s      r/   rq   zinvweibull_gen._ppf  s"    x
DF+++r1   c                 :    t          j        |            d|z  z  S )Nr  r  )r:   r}  r  s      r/   rt   zinvweibull_gen._isf  s    1"A&&r1   c                 6    t          j        d||z  z
            S rQ   r  r'  s      r/   r   zinvweibull_gen._munp  s    xAE	"""r1   c                 V    dt           z   t           |z  z   t          j        |          z
  S rQ   r  r)  s     r/   r   zinvweibull_gen._entropy  s"    x&1*$rvayy00r1   Nc                 d    |dn|}t          t          |                               ||          S )N)r   r  )r7   r5  r  )r:   r   r;   r  s      r/   r  zinvweibull_gen._fitstart  s3    vv4^T**44T4EEEr1   rA   )rv   rw   rx   ry   r   r  r  r^   re   rh   rl   rq   rt   r   r   r  r  r  s   @r/   r5  r5    s         : "4ME E E    ! ! !, , ,' ' '# # #1 1 1F F F F F F F F F Fr1   r5  
invweibullc                   >    e Zd ZdZej        Zd Zd Zd Z	d Z
d ZdS )johnsonsb_gena!  A Johnson SB continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsu

    Notes
    -----
    The probability density function for `johnsonsb` is:

    .. math::

        f(x, a, b) = \frac{b}{x(1-x)}  \phi(a + b \log \frac{x}{1-x} )

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`
    and :math:`x \in [0,1]`.  :math:`\phi` is the pdf of the normal
    distribution.

    `johnsonsb` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     |dk    ||k    z  S r:  rz   r|  s      r/   rV   zjohnsonsb_gen._argcheck  r  r1   c                     t          ddt          j         t          j        fd          }t          dddt          j        fd          }||gS Nr~   Fr   r   r   r[   rY  s      r/   r^   zjohnsonsb_gen._shape_info  r  r1   c           	      ~    t          ||t          j        |d|z
  z            z  z             }|dz  |d|z
  z  z  |z  S r  )r   rC   r   )r:   rd   r~   r   trms        r/   re   zjohnsonsb_gen._pdf  sF    AbfQAY////00ua1gs""r1   c           	      \    t          ||t          j        |d|z
  z            z  z             S r  r   rC   r   rc  s       r/   rh   zjohnsonsb_gen._cdf  s,    QrvaQi0000111r1   c                 b    ddt          j        d|z  t          |          |z
  z            z   z  S )Nr|   r   r  rC   r   r   r:   rp   r~   r   s       r/   rq   zjohnsonsb_gen._ppf  s0    a"&Yq\\A-=!>???@@r1   N)rv   rw   rx   ry   r   r  r  rV   r^   re   rh   rq   rz   r1   r/   rD  rD    sx         6 "4M" " "  
# # #
2 2 2A A A A Ar1   rD  	johnsonsbc                   0    e Zd ZdZd Zd Zd Zd Zd ZdS )johnsonsu_gena%  A Johnson SU continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsb

    Notes
    -----
    The probability density function for `johnsonsu` is:

    .. math::

        f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
                     \phi(a + b \log(x + \sqrt{x^2 + 1}))

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`.
    :math:`\phi` is the pdf of the normal distribution.

    `johnsonsu` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     |dk    ||k    z  S r:  rz   r|  s      r/   rV   zjohnsonsu_gen._argcheck  r  r1   c                     t          ddt          j         t          j        fd          }t          dddt          j        fd          }||gS rG  r[   rY  s      r/   r^   zjohnsonsu_gen._shape_info   r  r1   c                     ||z  }t          ||t          j        |t          j        |dz             z             z  z             }|dz  t          j        |dz             z  |z  S r  )r   rC   r   r   )r:   rd   r~   r   r#  rI  s         r/   re   zjohnsonsu_gen._pdf%  s_     qSAq272a4=='8 9 999::uRWRV__$S((r1   c                     t          ||t          j        |t          j        ||z  dz             z             z  z             S rQ   )r   rC   r   r   rc  s       r/   rh   zjohnsonsu_gen._cdf,  s;    QBGAaC!G,<,<(<!=!===>>>r1   c                 P    t          j        t          |          |z
  |z            S rA   )rC   sinhr   rN  s       r/   rq   zjohnsonsu_gen._ppf/  s"    w	!q(A-...r1   N)	rv   rw   rx   ry   rV   r^   re   rh   rq   rz   r1   r/   rQ  rQ    si         6" " "  
) ) )? ? ?/ / / / /r1   rQ  	johnsonsuc                       e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Ze eed          d                         ZdS )laplace_gena
  A Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `laplace` is

    .. math::

        f(x) = \frac{1}{2} \exp(-|x|)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zlaplace_gen._shape_infoJ  r   r1   Nc                 2    |                     dd|          S )Nr   r   r  )laplacer   s      r/   r   zlaplace_gen._rvsM  s    ##Aqt#444r1   c                 L    dt          j        t          |                     z  S Nr   )rC   r   r  r   s     r/   re   zlaplace_gen._pdfP  s    263q66'??""r1   c           	          t          j        d          5  t          j        |dk    ddt          j        |           z  z
  dt          j        |          z            cd d d            S # 1 swxY w Y   d S )Nr+  )r  r   r|   r   )rC   r.  r  r   r   s     r/   rh   zlaplace_gen._cdfT  s    [h''' 	H 	H8AE3RVQBZZ#7RVAYYGG	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	H 	Hs   AA++A/2A/c                 .    |                      |           S rA   rh   r   s     r/   rl   zlaplace_gen._sfX  s    yy!}}r1   c                     t          j        |dk    t          j        dd|z
  z             t          j        d|z                      S r   rC   r  r   r   s     r/   rq   zlaplace_gen._ppf\  s9    xC"&AaC//!126!A#;;???r1   c                 .    |                      |           S rA   rq   r   s     r/   rt   zlaplace_gen._isf_  s    		!}r1   c                     dS )N)r   rH   r   r  rz   r]   s    r/   r   zlaplace_gen._statsc  s    zr1   c                 0    t          j        d          dz   S r  r#  r]   s    r/   r   zlaplace_gen._entropyf  s    vayy{r1   z        This function uses explicit formulas for the maximum likelihood
        estimation of the Laplace distribution parameters, so the keyword
        arguments `loc`, `scale`, and `optimizer` are ignored.

r   c                     t          | |||          \  }}}|t          j        |          }|9t          j        t          j        ||z
                      t          |          z  }||fS rA   )r  rC   medianr  r  r  )r:   r   r;   r.   r   r   s         r/   r9   zlaplace_gen.fiti  sp     9t9=tE EdF <9T??D>fRVD4K0011SYY>FV|r1   r   )rv   rw   rx   ry   r^   r   re   rh   rl   rq   rt   r   r   r>   r	   r   r9   rz   r1   r/   rZ  rZ  6  s         &  5 5 5 5# # #H H H  @ @ @        6F G G G 	G G _
  r1   rZ  r]  c                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZdS )laplace_asymmetric_genu
  An asymmetric Laplace continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution

    Notes
    -----
    The probability density function for `laplace_asymmetric` is

    .. math::

       f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
                    &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\

    for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.

    `laplace_asymmetric` takes ``kappa`` as a shape parameter for
    :math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
    Laplace distribution.

    %(after_notes)s

    References
    ----------
    .. [1] "Asymmetric Laplace distribution", Wikipedia
            https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

    .. [2] Kozubowski TJ and Podgórski K. A Multivariate and
           Asymmetric Generalization of Laplace Distribution,
           Computational Statistics 15, 531--540 (2000).
           :doi:`10.1007/PL00022717`

    %(example)s

    c                 @    t          dddt          j        fd          gS NkappaFr   r   r[   r]   s    r/   r^   z"laplace_asymmetric_gen._shape_info      7EArv;GGHHr1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   ro  s      r/   re   zlaplace_asymmetric_gen._pdf  s     vdll1e,,---r1   c                     d|z  }|t          j        |dk    | |          z  }|t          j        ||z             z  }|S r  rd  )r:   rd   ro  kapinvrh  s        r/   r   zlaplace_asymmetric_gen._logpdf  sF    5"(16E66222rveFl###
r1   c                     d|z  }||z   }t          j        |dk    dt          j        | |z            ||z  z  z
  t          j        ||z            ||z  z            S r  rC   r  r   r:   rd   ro  rt  
kappkapinvs        r/   rh   zlaplace_asymmetric_gen._cdf  sk    56\
xQBFA2e8,,fZ.?@@qx((%
*:;= = 	=r1   c           	          d|z  }||z   }t          j        |dk    t          j        | |z            ||z  z  dt          j        ||z            ||z  z  z
            S r  rv  rw  s        r/   rl   zlaplace_asymmetric_gen._sf  sn    56\
xQr%x((&*;<BF1V8,,eJ.>??A A 	Ar1   c                     d|z  }||z   }t          j        |||z  k    t          j        d|z
  |z  |z             |z  t          j        ||z  |z            |z            S rQ   rd  r:   rp   ro  rt  rx  s        r/   rq   zlaplace_asymmetric_gen._ppf  sr    56\
xU:--Q
 25 8999&@q|E12258: : 	:r1   c                     d|z  }||z   }t          j        |||z  k    t          j        ||z  |z             |z  t          j        d|z
  |z  |z            |z            S rQ   rd  r{  s        r/   rt   zlaplace_asymmetric_gen._isf  su    56\
xVJ..*U 2333F:Az1%788>@ @ 	@r1   c                 T   d|z  }||z
  }||z  ||z  z   }ddt          j        |d          z
  z  t          j        dt          j        |d          z   d          z  }ddt          j        |d          z   z  t          j        dt          j        |d          z   d          z  }||||fS )	Nr   r   r  r  ri  r  r  rH   r3  )r:   ro  rt  mnr  r7  r8  s          r/   r   zlaplace_asymmetric_gen._stats  s    5e^VmeEk)!BHUA&&&'28E13E3E1Es(K(KK!BHUA&&&'28E13E3E1Eq(I(II3Br1   c                 <    dt          j        |d|z  z             z   S rQ   r#  r:   ro  s     r/   r   zlaplace_asymmetric_gen._entropy  s    26%%-((((r1   Nr  rz   r1   r/   rl  rl    s        % %LI I I. . .  = = =A A A: : :@ @ @  ) ) ) ) )r1   rl  laplace_asymmetricc                    t          j        |          }|                    dd           }|                    dd           }| j        r't	          | j                            d                    nd}g }g }| j        r| j                            dd                                          }	t          |	          D ]c\  }
}dt          |
          z   }|d|z   d|z   g}t          ||          }|
                    |           |
                    |           ||||<   ddd	d
dddh|}t          |                              |          }|rt          d| d          t	          |          |k    rt          d          d ||h|vrt          d          t          j        |                                          st#          d          |g|||R S )Nr   r   ,r    r  fix_r'   r(   r)   r*   zUnknown keyword arguments: r~  zToo many positional arguments.r   r   )rC   r   r5   shapesr  splitrG  	enumeratestrr   r  set
differencer-   r]  r   r   r   )distr   r;   r.   r   r   
num_shapesfshape_keysfshapesr  r  r  keynamesr  
known_keysunknown_keyss                    r/   r  r    s   :dD88FD!!DXXh%%F04BT[&&s++,,,JKG
 { 	 $$S#..4466f%% 	  	 DAqA,C#'6A:.E&tU33Cs###NN3S	 +x(2%02Jt99''
33L GElEEEFFF
4yy:8999D&+7+++  ' ( ( 	( ;t  "" A?@@@)7)D)&)))r1   c                   J    e Zd ZdZej        Zd Zd Zd Z	d Z
d Zd Zd Zd	S )
levy_genah  A Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy_stable, levy_l

    Notes
    -----
    The probability density function for `levy` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)

    for :math:`x >= 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the upper 40 percent.
    >>> a, b = levy.ppf(0), levy.ppf(0.6)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)]))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     g S rA   rz   r]   s    r/   r^   zlevy_gen._shape_infoX  r   r1   c                     dt          j        dt           j        z  |z            z  |z  t          j        dd|z  z            z  S Nr   rH   r  rQ  r   s     r/   re   zlevy_gen._pdf[  s=    271RU719%%%)BF2qs8,<,<<<r1   c                 T    t          j        t          j        d|z                      S r_  )rj   erfcrC   r   r   s     r/   rh   zlevy_gen._cdf_  s     wrwsQw''(((r1   c                 T    t          j        t          j        d|z                      S r_  )rj   rr  rC   r   r   s     r/   rl   zlevy_gen._sfc  s     vbgcAg&&'''r1   c                 B    t          j        |dz             }d||z  z  S rU  r   r:   rp   r  s      r/   rq   zlevy_gen._ppff  s$    x!}}ncCi  r1   c                 <    ddt          j        |          dz  z  z  S r"  )rj   erfinvr  s     r/   rt   zlevy_gen._isfk  s    !BIaLL!O#$$r1   c                 ^    t           j        t           j        t           j        t           j        fS rA   r
  r]   s    r/   r   zlevy_gen._statsn  r=  r1   Nrv   rw   rx   ry   r   r  r  r^   re   rh   rl   rq   rt   r   rz   r1   r/   r  r    s        G GP "4M  = = =) ) )( ( (! ! !
% % %. . . . .r1   r  levyc                   J    e Zd ZdZej        Zd Zd Zd Z	d Z
d Zd Zd Zd	S )

levy_l_gena  A left-skewed Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy, levy_stable

    Notes
    -----
    The probability density function for `levy_l` is:

    .. math::
        f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}

    for :math:`x <= 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=-1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy_l
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy_l.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy_l` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the lower 40 percent.
    >>> a, b = levy_l.ppf(0.4), levy_l.ppf(1)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy_l.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy_l pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy_l()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy_l.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy_l.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20)))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     g S rA   rz   r]   s    r/   r^   zlevy_l_gen._shape_info  r   r1   c                     t          |          }dt          j        dt          j        z  |z            z  |z  t          j        dd|z  z            z  S r  )r  rC   r   r   r   r:   rd   r  s      r/   re   zlevy_l_gen._pdf  sH    VV25$$$R'r1R4y(9(999r1   c                 t    t          |          }dt          dt          j        |          z            z  dz
  S r  )r  r   rC   r   r  s      r/   rh   zlevy_l_gen._cdf  s1    VV9Q_---11r1   c                 n    t          |          }dt          dt          j        |          z            z  S r  )r  r   rC   r   r  s      r/   rl   zlevy_l_gen._sf  s,    VV8AO,,,,r1   c                 <    t          |dz   dz            }d||z  z  S )Nr|   rH   r  r   r  s      r/   rq   zlevy_l_gen._ppf  s&    SA&&sSy!!r1   c                 2    dt          |dz            dz  z  S )Nr  rH   r   r  s     r/   rt   zlevy_l_gen._isf  s    )AaC..!###r1   c                 ^    t           j        t           j        t           j        t           j        fS rA   r
  r]   s    r/   r   zlevy_l_gen._stats  r=  r1   Nr  rz   r1   r/   r  r  u  s        F FN "4M  : : :
2 2 2- - -" " "$ $ $. . . . .r1   r  levy_lc                        e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd Zd Zd Ze ee           fd                        Z xZS )logistic_gena  A logistic (or Sech-squared) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `logistic` is:

    .. math::

        f(x) = \frac{\exp(-x)}
                    {(1+\exp(-x))^2}

    `logistic` is a special case of `genlogistic` with ``c=1``.

    Remark that the survival function (``logistic.sf``) is equal to the
    Fermi-Dirac distribution describing fermionic statistics.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zlogistic_gen._shape_info  r   r1   Nc                 .    |                     |          S rO  )logisticr   s      r/   r   zlogistic_gen._rvs  s    $$$$///r1   c                 P    t          j        |                     |                    S rA   r  r   s     r/   re   zlogistic_gen._pdf  r  r1   c                     t          j        |           }|dt          j        t          j        |                    z  z
  S r0  )rC   r  rj   r  r   )r:   rd   rV  s      r/   r   zlogistic_gen._logpdf  s3    VAYYJ2+++++r1   c                 *    t          j        |          S rA   rj   expitr   s     r/   rh   zlogistic_gen._cdf      x{{r1   c                 *    t          j        |          S rA   rj   	log_expitr   s     r/   r   zlogistic_gen._logcdf  s    |Ar1   c                 *    t          j        |          S rA   rj   logitr   s     r/   rq   zlogistic_gen._ppf	  r  r1   c                 ,    t          j        |           S rA   r  r   s     r/   rl   zlogistic_gen._sf  s    x||r1   c                 ,    t          j        |           S rA   r  r   s     r/   r   zlogistic_gen._logsf  s    |QBr1   c                 ,    t          j        |           S rA   r  r   s     r/   rt   zlogistic_gen._isf  s    |r1   c                 B    dt           j        t           j        z  dz  ddfS )Nr   r  g333333?r   r]   s    r/   r   zlogistic_gen._stats  s    "%+c/1g--r1   c                     dS r0  rz   r]   s    r/   r   zlogistic_gen._entropy  s    sr1   c                   
 |                     dd          r t                      j        g|R i |S t          | ||          \  }}t	                    |                               \  }}|                    d|          |                    d|          }}|ffd	
|ffd	
fd}|(|&t          j        
|f          }	|	j	        d         }|}nK|(|&t          j        |f          }	|	j	        d         }|}n!t          j        |||f          }	|	j	        \  }}|	j
        r||fn t                      j        g|R i |S )	Nr  Fr'   r(   c                 l    | z
  |z  }t          j        t          j        |                    dz  z
  S r	  )rC   r  rj   r  )r'   r(   r  r   rU   s      r/   dl_dlocz!logistic_gen.fit.<locals>.dl_dloc/  s2    u$A6"(1++&&1,,r1   c                 r    |z
  | z  }t          j        |t          j        |dz            z            z
  S r	  )rC   r  rD  )r(   r'   r  r   rU   s      r/   	dl_dscalez#logistic_gen.fit.<locals>.dl_dscale3  s6    u$A6!BGAaCLL.))A--r1   c                 >    | \  }} ||           ||          fS rA   rz   )paramsr'   r(   r  r  s      r/   rO  zlogistic_gen.fit.<locals>.func7  s/    JC73&&		%(=(===r1   r   )r,   r7   r9   r  r  r  r5   r   r  rd   success)r:   r   r;   r.   r   r   r'   r(   rO  r  r  r  rU   r  s    `        @@@r/   r9   zlogistic_gen.fit  s    88J&& 	4577;t3d333d3338t9=tE EdFII ^^D))
UXXeS))488GU+C+CU  & 	- 	- 	- 	- 	- 	- 	- "& 	. 	. 	. 	. 	. 	. 	.	> 	> 	> 	> 	> 	> $,-#00C%(CEE&.-	E844CE!HECC-sEl33CJC # 6e UWW[555555	7r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   r   rq   rl   r   rt   r   r   r>   r
   r   r9   r  r  s   @r/   r  r    s        .  0 0 0 0' ' ', , ,               . . .   M***7 *7 *7 *7 +* _*7 *7 *7 *7 *7r1   r  r  c                   J    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd ZdS )loggamma_gena  A log gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loggamma` is:

    .. math::

        f(x, c) = \frac{\exp(c x - \exp(x))}
                       {\Gamma(c)}

    for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `loggamma` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zloggamma_gen._shape_infog  r   r1   Nc                     t          j        |                    |dz   |                    t          j        |                    |                    |z  z   S )Nr   r  )rC   r   r  r  rA  s       r/   r   zloggamma_gen._rvsj  sU     |))!a%d);;<<&--4-8899!;< 	=r1   c                     t          j        ||z  t          j        |          z
  t          j        |          z
            S rA   rC   r   rj   rW  r  s      r/   re   zloggamma_gen._pdfx  s/    vac"&))mBJqMM1222r1   c                 `    ||z  t          j        |          z
  t          j        |          z
  S rA   r  r  s      r/   r   zloggamma_gen._logpdf|  s%    sRVAYYA..r1   c                 P    t          j        |t          j        |                    S rA   )rj   r\  rC   r   r  s      r/   rh   zloggamma_gen._cdf  s    {1bfQii(((r1   c                 P    t          j        t          j        ||                    S rA   )rC   r   rj   rc  r  s      r/   rq   zloggamma_gen._ppf  s    vbnQ**+++r1   c                 P    t          j        |t          j        |                    S rA   )rj   r_  rC   r   r  s      r/   rl   zloggamma_gen._sf  s    |Arvayy)))r1   c                 P    t          j        t          j        ||                    S rA   )rC   r   rj   rg  r  s      r/   rt   zloggamma_gen._isf  s    vboa++,,,r1   c                     t          j        |          }t          j        d|          }t          j        d|          t          j        |d          z  }t          j        d|          ||z  z  }||||fS )Nr   rH   ri  r  )rj   rT  	polygammarC   rj  )r:   r  r   r  r1  excess_kurtosiss         r/   r   zloggamma_gen._stats  sm     z!}}l1a  <1%%c(:(::,q!,,C8S(O33r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rq   rl   rt   r   rz   r1   r/   r  r  N  s         0E E E= = = =3 3 3/ / /) ) ), , ,* * *- - -4 4 4 4 4r1   r  loggammac                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	loglaplace_gena|  A log-Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loglaplace` is:

    .. math::

        f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1}  &\text{for } 0 < x < 1\\
                               \frac{c}{2} x^{-c-1}  &\text{for } x \ge 1
                  \end{cases}

    for :math:`c > 0`.

    `loglaplace` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
    The Mathematical Scientist, vol. 28, pp. 49-60, 2003.

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zloglaplace_gen._shape_info  r   r1   c                 X    |dz  }t          j        |dk     ||           }|||dz
  z  z  S Nr   r   rC   r  )r:   rd   r  cd2s       r/   re   zloglaplace_gen._pdf  s8     eHQUAr""1qs8|r1   c                 V    t          j        |dk     d||z  z  dd|| z  z  z
            S Nr   r   r  r  s      r/   rh   zloglaplace_gen._cdf  s0    xAs1a4x3qA2w;777r1   c                 `    t          j        |dk     d|z  d|z  z  dd|z
  z  d|z  z            S )Nr   r   r|   rH   r  r  r  s      r/   rq   zloglaplace_gen._ppf  s8    xC#a%3q5!1As1uIa3HIIIr1   c                 $    |dz  |dz  |dz  z
  z  S r	  rz   r'  s      r/   r   zloglaplace_gen._munp  s    !tq!tad{##r1   c                 6    t          j        d|z            dz   S r  r#  r)  s     r/   r   zloglaplace_gen._entropy  s    vc!e}}s""r1   N)
rv   rw   rx   ry   r^   re   rh   rq   r   r   rz   r1   r/   r  r    s~         8E E E  8 8 8J J J$ $ $# # # # #r1   r  
loglaplacec                 J    t          | dk    | |fd t          j                   S )Nr   c                     t          j        |           dz   d|dz  z  z  t          j        || z  t          j        dt           j        z            z            z
  S r	  )rC   r   r   r   rd   r  s     r/   r  z!_lognorm_logpdf.<locals>.<lambda>  sM    BF1IIqL=AadF#;bfQqSQRSUSXQXIYIYEY>Z>Z#Z r1   r  r  s     r/   _lognorm_logpdfr    s-    a1fq!fZZvg  r1   c                        e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zd Zd Ze eed           fd                        Z xZS )lognorm_genab  A lognormal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lognorm` is:

    .. math::

        f(x, s) = \frac{1}{s x \sqrt{2\pi}}
                  \exp\left(-\frac{\log^2(x)}{2s^2}\right)

    for :math:`x > 0`, :math:`s > 0`.

    `lognorm` takes ``s`` as a shape parameter for :math:`s`.

    %(after_notes)s

    Suppose a normally distributed random variable ``X`` has  mean ``mu`` and
    standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally
    distributed with ``s = sigma`` and ``scale = exp(mu)``.

    %(example)s

    c                 @    t          dddt          j        fd          gS )Nr  Fr   r   r[   r]   s    r/   r^   zlognorm_gen._shape_info  r   r1   Nc                 V    t          j        ||                    |          z            S rA   rC   r   r   )r:   r  r   r   s       r/   r   zlognorm_gen._rvs  s%    va,66t<<<===r1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   r  s      r/   re   zlognorm_gen._pdf  r  r1   c                 "    t          ||          S rA   r  r  s      r/   r   zlognorm_gen._logpdf  s    q!$$$r1   c                 J    t          t          j        |          |z            S rA   rK  r  s      r/   rh   zlognorm_gen._cdf  s    Q'''r1   c                 J    t          t          j        |          |z            S rA   r  r  s      r/   r   zlognorm_gen._logcdf  s    BF1IIM***r1   c                 J    t          j        |t          |          z            S rA   rM  )r:   rp   r  s      r/   rq   zlognorm_gen._ppf  s    va)A,,&'''r1   c                 J    t          t          j        |          |z            S rA   )r   rC   r   r  s      r/   rl   zlognorm_gen._sf  s    q		A&&&r1   c                 J    t          t          j        |          |z            S rA   )r   rC   r   r  s      r/   r   zlognorm_gen._logsf  s    26!99q=)))r1   c                     t          j        ||z            }t          j        |          }||dz
  z  }t          j        |dz
            d|z   z  }t          j        g d|          }||||fS Nr   rH   )r   rH   r  r   r  )rC   r   r   polyval)r:   r  r}  r5  r6  r7  r8  s          r/   r   zlognorm_gen._stats  sm    F1Q3KKWQZZ1gWac^^QqS!Z***A..3Br1   c                     ddt          j        dt           j        z            z   dt          j        |          z  z   z  S )Nr   r   rH   r   )r:   r  s     r/   r   zlognorm_gen._entropy  s1    a"&25//)Aq		M9::r1   aF          When `method='MLE'` and
        the location parameter is fixed by using the `floc` argument,
        this function uses explicit formulas for the maximum likelihood
        estimation of the log-normal shape and scale parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are ignored.
        

r   c                 ~   |                     dd           }| t                      j        |g|R i |S |                     dd           p+|                     dd           p|                     dd           }|                     dd           }t          |          dk    rt	          d          dD ]}|                    |d            |rt	          d	|z            ||t          d
          t          j        |          }t          j	        |          
                                st          d          t          |          }|dk    r||z
  }t          j        |dk              rt          d|t          j                  t          j        |          }|Mt          j        |                                          }	||                                }
n]t          |          }
nMt          |          }	t          j        |t          j        |	          z
  dz                                            }
|
||	fS )Nr   r  fsfix_sr   r   zToo many input arguments.)	r  r  r   r   r   r'   r(   r)   r*   r+   r   r   r   lognormr  rH   )r5   r7   r9   r  r-   r,   r   rC   r   r   r   r  r  r<  r\   r   r   r   stdr   )r:   r   r;   r.   r   r  r   r   lndatar(   r  r  s              r/   r9   zlognorm_gen.fit  s=    xx%%<577;t3d333d333hhtT"" &dhhtT&:&: &hhw%% 	(D))t99q==7888, 	! 	!DHHT4     	=4t;<<<
 >f0 ) * * * z${4  $$&& 	ECDDDT{{199 $;D6$!) 	DyBFCCCC >F6;;==))Ez

 b		 &MMEGfrve}}4q8>>@@AAEdE!!r1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   rh   r   rq   rl   r   r   r   r>   r   r9   r  r  s   @r/   r  r    s+        4 "4ME E E> > > >* * *% % %( ( (+ + +( ( (' ' '* * *  ; ; ; } 5   ;" ;" ;" ;"  _;" ;" ;" ;" ;"r1   r  r  c                   R    e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 ZdS )
gibrat_genaB  A Gibrat continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gibrat` is:

    .. math::

        f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)

    `gibrat` is a special case of `lognorm` with ``s=1``.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zgibrat_gen._shape_infox  r   r1   Nc                 P    t          j        |                    |                    S rA   r  r   s      r/   r   zgibrat_gen._rvs{  s     vl22488999r1   c                 P    t          j        |                     |                    S rA   r  r   s     r/   re   zgibrat_gen._pdf~  r  r1   c                 "    t          |d          S r  r  r   s     r/   r   zgibrat_gen._logpdf  s    q#&&&r1   c                 D    t          t          j        |                    S rA   rK  r   s     r/   rh   zgibrat_gen._cdf  s    ###r1   c                 D    t          j        t          |                    S rA   rM  r   s     r/   rq   zgibrat_gen._ppf  s    vill###r1   c                     t           j        }t          j        |          }||dz
  z  }t          j        |dz
            d|z   z  }t          j        g d|          }||||fS r  )rC   er   r  )r:   r}  r5  r6  r7  r8  s         r/   r   zgibrat_gen._stats  se    DWQZZ1q5kWa!eQ'Z***A..3Br1   c                 P    dt          j        dt           j        z            z  dz   S rZ  r   r]   s    r/   r   zgibrat_gen._entropy  s"    RVAI&&&,,r1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   rh   rq   r   r   rz   r1   r/   r  r  b  s         & "4M  : : : :' ' '' ' '$ $ $$ $ $  - - - - -r1   r  zp`gilbrat` is a misspelling of the correct name for the `gibrat` distribution, and will be removed in SciPy 1.11.c                       e Zd ZdZd ZdS )gilbrat_genz

    .. deprecated:: 1.9.0
        `gilbrat` is deprecated, use `gibrat` instead!
        `gilbrat` is a misspelling of the correct name for the `gibrat`
        distribution, and will be removed in SciPy 1.11.

    c                 h    dt           z   }t          j        |t          d            | j        |i |S )Nz/`gilbrat` is deprecated, use `gibrat` instead!
rH   
stacklevel)deprmsgrp  r  DeprecationWarningfreeze)r:   r;   r.   rL   s       r/   __call__zgilbrat_gen.__call__  s;    @7Jc-!<<<<t{D)D)))r1   N)rv   rw   rx   ry   r  rz   r1   r/   r  r    s-         * * * * *r1   r  gibratgilbrat)r   entropyexpectr9   intervalr  logcdfr-  logsfr   rj  momentpdfr  r  sfr  r  r  zgilbrat.zgibrat.c                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )maxwell_gena  A Maxwell continuous random variable.

    %(before_notes)s

    Notes
    -----
    A special case of a `chi` distribution,  with ``df=3``, ``loc=0.0``,
    and given ``scale = a``, where ``a`` is the parameter used in the
    Mathworld description [1]_.

    The probability density function for `maxwell` is:

    .. math::

        f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)

    for :math:`x >= 0`.

    %(after_notes)s

    References
    ----------
    .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html

    %(example)s
    c                     g S rA   rz   r]   s    r/   r^   zmaxwell_gen._shape_info  r   r1   Nc                 <    t                               d||          S )Nr  r  rl  r  r   s      r/   r   zmaxwell_gen._rvs  s    wwsLwAAAr1   c                 T    t           |z  |z  t          j        | |z  dz            z  S r0  )r"   rC   r   r   s     r/   re   zmaxwell_gen._pdf  s+    q "261"Q$s(#3#333r1   c                     t          j        d          5  t          dt          j        |          z  z   d|z  |z  z
  cd d d            S # 1 swxY w Y   d S )Nr+  r,  rH   r   )rC   r.  r#   r   r   s     r/   r   zmaxwell_gen._logpdf  s    [))) 	? 	?&26!994s1uQw>	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	? 	?s   (AAAc                 8    t          j        d||z  dz            S )Nri  r   r[  r   s     r/   rh   zmaxwell_gen._cdf  s    {3!C(((r1   c                 V    t          j        dt          j        d|          z            S r<  rb  r   s     r/   rq   zmaxwell_gen._ppf  s#    wqQ///000r1   c                 R   dt           j        z  dz
  }dt          j        dt           j        z            z  ddt           j        z  z
  t          j        d          ddt           j        z  z
  z  |dz  z  dt           j        z  t           j        z  d	t           j        z  z   d
z
  |dz  z  fS )Nr  r  rH   r       rW  ri  r-     i  rC   r   r   r:   r  s     r/   r   zmaxwell_gen._stats  s    gai"'#be)$$$!BE'	

Br"%xK(c1RU253ru9,s2c3h>@ 	@r1   c                 `    t           dt          j        dt          j        z            z  z   dz
  S rZ  )r   rC   r   r   r]   s    r/   r   zmaxwell_gen._entropy  s%    BF1RU7OO++C//r1   r   rZ  rz   r1   r/   r#  r#    s         4  B B B B4 4 4? ? ?
) ) )1 1 1@ @ @0 0 0 0 0r1   r#  maxwellc                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	
mielke_gena  A Mielke Beta-Kappa / Dagum continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `mielke` is:

    .. math::

        f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}

    for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
    called Dagum distribution ([2]_). It was already defined in [3]_, called
    a Burr Type III distribution (`burr` with parameters ``c=s`` and
    ``d=k/s``).

    `mielke` takes ``k`` and ``s`` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing
           and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280
    .. [2] Dagum, C., 1977 "A new model for personal income distribution."
           Economie Appliquee, 33, 327-367.
    .. [3] Burr, I. W. "Cumulative frequency functions", Annals of
           Mathematical Statistics, 13(2), pp 215-232 (1942).

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )Nr  Fr   r   r  r[   )r:   iki_ss      r/   r^   zmielke_gen._shape_info   >    UQK@@ea[.AACyr1   c                 B    |||dz
  z  z  d||z  z   d|dz  |z  z   z  z  S r  rz   r:   rd   r  r  s       r/   re   zmielke_gen._pdf%  s2    QsU|s1a4x3quQw;777r1   c                     t          j        d          5  t          j        |          t          j        |          |dz
  z  z   t          j        ||z            d||z  z   z  z
  cd d d            S # 1 swxY w Y   d S )Nr+  r,  r   )rC   r.  r   r  r9  s       r/   r   zmielke_gen._logpdf(  s    [))) 	L 	L6!99rvayy!a%0028AqD>>1qs73KK	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	L 	Ls   AA33A7:A7c                 0    ||z  d||z  z   |dz  |z  z  z  S r  rz   r9  s       r/   rh   zmielke_gen._cdf-  s&    !ts1a4x1S57+++r1   c                 `    t          ||dz  |z            }t          |d|z
  z  d|z            S r  r  )r:   rp   r  r  qsks        r/   rq   zmielke_gen._ppf0  s3    !QsU1Woo3C=#a%(((r1   c                 N    d }t          ||k     |||f|t          j                  S )Nc                     t          j        || z   |z            t          j        d| |z  z
            z  t          j        ||z            z  S rQ   r  )rU   r  r  s      r/   
nth_momentz$mielke_gen._munp.<locals>.nth_moment5  s@    8QqS!G$$RXa!e__4RXac]]BBr1   r  )r:   rU   r  r  r@  s        r/   r   zmielke_gen._munp4  s6    	C 	C 	C !a%!QJ???r1   N)
rv   rw   rx   ry   r^   re   r   rh   rq   r   rz   r1   r/   r3  r3    s           B  
8 8 8L L L
, , ,) ) )@ @ @ @ @r1   r3  mielkec                   T    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd ZdS )
kappa4_genap  Kappa 4 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for kappa4 is:

    .. math::

        f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}

    if :math:`h` and :math:`k` are not equal to 0.

    If :math:`h` or :math:`k` are zero then the pdf can be simplified:

    h = 0 and k != 0::

        kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                              exp(-(1.0 - k*x)**(1.0/k))

    h != 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

    h = 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

    kappa4 takes :math:`h` and :math:`k` as shape parameters.

    The kappa4 distribution returns other distributions when certain
    :math:`h` and :math:`k` values are used.

    +------+-------------+----------------+------------------+
    | h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
    +======+=============+================+==================+
    | -1.0 | Logistic    |                | Generalized      |
    |      |             |                | Logistic(1)      |
    |      |             |                |                  |
    |      | logistic(x) |                |                  |
    +------+-------------+----------------+------------------+
    |  0.0 | Gumbel      | Reverse        | Generalized      |
    |      |             | Exponential(2) | Extreme Value    |
    |      |             |                |                  |
    |      | gumbel_r(x) |                | genextreme(x, k) |
    +------+-------------+----------------+------------------+
    |  1.0 | Exponential | Uniform        | Generalized      |
    |      |             |                | Pareto           |
    |      |             |                |                  |
    |      | expon(x)    | uniform(x)     | genpareto(x, -k) |
    +------+-------------+----------------+------------------+

    (1) There are at least five generalized logistic distributions.
        Four are described here:
        https://en.wikipedia.org/wiki/Generalized_logistic_distribution
        The "fifth" one is the one kappa4 should match which currently
        isn't implemented in scipy:
        https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
        https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
    (2) This distribution is currently not in scipy.

    References
    ----------
    J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
    to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
    Faculty of the Louisiana State University and Agricultural and Mechanical
    College, (August, 2004),
    https://digitalcommons.lsu.edu/gradschool_dissertations/3672

    J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
    Develop. 38 (3), 25 1-258 (1994).

    B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
    Site in the Chi River Basin, Thailand", Journal of Water Resource and
    Protection, vol. 4, 866-869, (2012).
    :doi:`10.4236/jwarp.2012.410101`

    C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
    Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
    2000).
    http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf

    %(after_notes)s

    %(example)s

    c                 n    t          j        ||          d         j        }t          j        |d          S )Nr   T
fill_value)rC   r  r  full)r:   r  r  r  s       r/   rV   zkappa4_gen._argcheck  s1    #Aq))!,2wu....r1   c                     t          ddt          j         t          j        fd          }t          ddt          j         t          j        fd          }||gS )Nr  Fr   r  r[   )r:   ihr5  s      r/   r^   zkappa4_gen._shape_info  sG    UbfWbf$5~FFUbfWbf$5~FFBxr1   c           
         t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk               t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk               g}d }d }d }d }t          |||||||g||gt           j                  }d }d }t          |||||||g||gt           j                  }	||	fS )	Nr   c                 :    dt          j        | |           z
  |z  S r  )rC   r  r  r  s     r/   r  z#kappa4_gen._get_support.<locals>.f0  s     ".QB///22r1   c                 *    t          j        |           S rA   r#  rL  s     r/   r  z#kappa4_gen._get_support.<locals>.f1  s    6!99r1   c                 v    t          j        t          j        |                     }t           j         |d d <   |S rA   rC   r  r  r\   r  r  r~   s      r/   f3z#kappa4_gen._get_support.<locals>.f3  s/    !%%AF7AaaaDHr1   c                     d|z  S r  rz   rL  s     r/   f5z#kappa4_gen._get_support.<locals>.f5      q5Lr1   defaultc                     d|z  S r  rz   rL  s     r/   r  z#kappa4_gen._get_support.<locals>.f0  rT  r1   c                 t    t          j        t          j        |                     }t           j        |d d <   |S rA   rO  rP  s      r/   r  z#kappa4_gen._get_support.<locals>.f1  s-    !%%A6AaaaDHr1   rC   r  r   r  )
r:   r  r  condlistr  r  rQ  rS  r  r  s
             r/   r   zkappa4_gen._get_support  s`   N1q5!a%00N1q5!q&11N1q5!a%00N161q511N161622N161q5113	3 	3 	3	 	 		 	 	
	 	 	  "b"b"5V%'V- - -
	 	 		 	 	
  "b"b"5V%'V- - - 2vr1   c                 T    t          j        |                     |||                    S rA   r  r:   rd   r  r  s       r/   re   zkappa4_gen._pdf  r   r1   c                 F   t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              g}d }d }d }d }t          |||||g|||gt           j                  S )Nr   c                     t          j        d|z  dz
  | | z            t          j        d|z  dz
  | d|| z  z
  d|z  z  z            z   S )zpdf = (1.0 - k*x)**(1.0/k - 1.0)*(
                      1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
               logpdf = ...
            r|   r  rd   r  r  s      r/   r  zkappa4_gen._logpdf.<locals>.f0  s\    
 Js1us{QBqD11Js1us{QBac	SU/C,CDDE Fr1   c                 ^    t          j        d|z  dz
  | | z            d|| z  z
  d|z  z  z
  S )z~pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
                      1.0 - k*x)**(1.0/k))
               logpdf = ...
            r|   r  r_  s      r/   r  zkappa4_gen._logpdf.<locals>.f1  s:    
 :c!eckA2a400C!A#IQ3GGGr1   c                 n    |  t          j        d|z  dz
  | t          j        |            z            z   S )z]pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
               logpdf = ...
            r|   )rj   re  rC   r   r_  s      r/   r  zkappa4_gen._logpdf.<locals>.f2  s5     2
3q53;261"::>>>>r1   c                 4    |  t          j        |            z
  S )zDpdf = np.exp(-x-np.exp(-x))
               logpdf = ...
            r  r_  s      r/   rQ  zkappa4_gen._logpdf.<locals>.f3  s     2r

?"r1   rU  rY  	r:   rd   r  r  rZ  r  r  r  rQ  s	            r/   r   zkappa4_gen._logpdf  s    N161622N161622N161622N1616224
	F 	F 	F	H 	H 	H	? 	? 	?	# 	# 	# 8B+q!9#%6+ + + 	+r1   c                 T    t          j        |                     |||                    S rA   r0  r\  s       r/   rh   zkappa4_gen._cdf  r  r1   c                 F   t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              g}d }d }d }d }t          |||||g|||gt           j                  S )Nr   c                 V    d|z  t          j        | d|| z  z
  d|z  z  z            z  S )zVcdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
               logcdf = ...
            r|   r  r_  s      r/   r  zkappa4_gen._logcdf.<locals>.f0  s5     E28QBac	SU';$;<<<<r1   c                      d|| z  z
  d|z  z   S )zLcdf = np.exp(-(1.0 - k*x)**(1.0/k))
               logcdf = ...
            r|   rz   r_  s      r/   r  zkappa4_gen._logcdf.<locals>.f1  s     1Q3Y#a%(((r1   c                 d    d|z  t          j        | t          j        |            z            z  S )zLcdf = (1.0 - h*np.exp(-x))**(1.0/h)
               logcdf = ...
            r|   )rj   r  rC   r   r_  s      r/   r  zkappa4_gen._logcdf.<locals>.f2
  s-     E28QBrvqbzzM2222r1   c                 .    t          j        |             S )zBcdf = np.exp(-np.exp(-x))
               logcdf = ...
            r  r_  s      r/   rQ  zkappa4_gen._logcdf.<locals>.f3  s     FA2JJ;r1   rU  rY  rc  s	            r/   r   zkappa4_gen._logcdf  s    N161622N161622N161622N1616224
	= 	= 	=	) 	) 	)	3 	3 	3	 	 	 8B+q!9#%6+ + + 	+r1   c                 F   t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              t          j        |dk    |dk              g}d }d }d }d }t          |||||g|||gt           j                  S )Nr   c                 0    d|z  dd| |z  z
  |z  |z  z
  z  S r  rz   rp   r  r  s      r/   r  zkappa4_gen._ppf.<locals>.f0!  s(    q5##A,!1A 5566r1   c                 D    d|z  dt          j        |            |z  z
  z  S r  r#  rl  s      r/   r  zkappa4_gen._ppf.<locals>.f1$  s$    q5#"&))a/00r1   c                 ^    t          j        | |z              t          j        |          z   S )z,ppf = -np.log((1.0 - (q**h))/h)
            r  rl  s      r/   r  zkappa4_gen._ppf.<locals>.f2'  s*     Hq!tW%%%q		11r1   c                 R    t          j        t          j        |                       S rA   r#  rl  s      r/   rQ  zkappa4_gen._ppf.<locals>.f3,  s    FBF1II:&&&&r1   rU  rY  )	r:   rp   r  r  rZ  r  r  r  rQ  s	            r/   rq   zkappa4_gen._ppf  s    N161622N161622N161622N1616224
	7 	7 	7	1 	1 	1	2 	2 	2
	' 	' 	' 8B+q!9#%6+ + + 	+r1   c                     t          j        |dk     |dk              |dk     g}d }d }t          |||g||gd          S )Nr   c                 B    d| z  |z                       t                    S r  astyper  rL  s     r/   r  z&kappa4_gen._get_stats_info.<locals>.f0:  s    F1H$$S)))r1   c                 <    d|z                       t                    S r  rr  rL  s     r/   r  z&kappa4_gen._get_stats_info.<locals>.f1=  s    F??3'''r1   r  rU  )rC   r  r   )r:   r  r  rZ  r  r  s         r/   _get_stats_infozkappa4_gen._get_stats_info4  sf    N1q5!q&))E

	* 	* 	*	( 	( 	( 8b"X1vqAAAAr1   c                 |    |                      ||          fdt          dd          D             }|d d          S )Nc                 \    g | ](}t          j        |k               rd nt           j        )S rA   rC   r  r  )r  r  maxrs     r/   r  z%kappa4_gen._stats.<locals>.<listcomp>D  s2    MMMA26!d(++744MMMr1   r   r  )ru  r  )r:   r  r  outputsry  s       @r/   r   zkappa4_gen._statsB  sG    ##Aq))MMMMq!MMMqqqzr1   c                     |                      |d         |d                   }||k    rt          j        S t          j        | j        dd|f|z             d         S Nr   r   r  )ru  rC   r  r   r  _mom_integ1)r:   rT  r;   ry  s       r/   _mom1_sczkappa4_gen._mom1_scG  sV    ##DGT!W55996M~d.1A49EEEaHHr1   N)rv   rw   rx   ry   rV   r^   r   re   r   rh   r   rq   ru  r   r~  rz   r1   r/   rC  rC  ?  s        W Wp/ / /  
' ' 'R- - -
$+ $+ $+L- - -!+ !+ !+F+ + +2B B B  
I I I I Ir1   rC  kappa4c                   6    e Zd ZdZd Zd Zd Zd Zd Zd Z	dS )	
kappa3_gena*  Kappa 3 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `kappa3` is:

    .. math::

        f(x, a) = a (a + x^a)^{-(a + 1)/a}

    for :math:`x > 0` and :math:`a > 0`.

    `kappa3` takes ``a`` as a shape parameter for :math:`a`.

    References
    ----------
    P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
    Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
    701-707, (September, 1973),
    :doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2`

    B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
    Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
    415-419 (2012), :doi:`10.4236/ojs.2012.24050`

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r   r[   r]   s    r/   r^   zkappa3_gen._shape_infor  r   r1   c                 *    ||||z  z   d|z  dz
  z  z  S r  rz   r  s      r/   re   zkappa3_gen._pdfu  s"    !ad(d1fQh'''r1   c                 $    ||||z  z   d|z  z  z  S r  rz   r  s      r/   rh   zkappa3_gen._cdfy  s    !ad(d1f%%%r1   c                 &    ||| z  dz
  z  d|z  z  S r  rz   r  s      r/   rq   zkappa3_gen._ppf|  s    1qb53;3q5))r1   c                 P    fdt          dd          D             }|d d          S )Nc                 \    g | ](}t          j        |k               rd nt           j        )S rA   rx  )r  r  r~   s     r/   r  z%kappa3_gen._stats.<locals>.<listcomp>  s0    JJJ26!a%==444bfJJJr1   r   r  )r  )r:   r~   rz  s    ` r/   r   zkappa3_gen._stats  s2    JJJJeAqkkJJJqqqzr1   c                     t          j        ||d         k              rt           j        S t          j        | j        dd|f|z             d         S r|  )rC   r  r  r   r  r}  )r:   rT  r;   s      r/   r~  zkappa3_gen._mom1_sc  sJ    6!tAw, 	6M~d.1A49EEEaHHr1   N)
rv   rw   rx   ry   r^   re   rh   rq   r   r~  rz   r1   r/   r  r  Q  s         @E E E( ( (& & &* * *  I I I I Ir1   r  kappa3c                   D    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 ZdS )	moyal_gena  A Moyal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `moyal` is:

    .. math::

        f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}

    for a real number :math:`x`.

    %(after_notes)s

    This distribution has utility in high-energy physics and radiation
    detection. It describes the energy loss of a charged relativistic
    particle due to ionization of the medium [1]_. It also provides an
    approximation for the Landau distribution. For an in depth description
    see [2]_. For additional description, see [3]_.

    References
    ----------
    .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
           The London, Edinburgh, and Dublin Philosophical Magazine
           and Journal of Science, vol 46, 263-280, (1955).
           :doi:`10.1080/14786440308521076` (gated)
    .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
           International Journal of Research and Reviews in Applied Sciences,
           vol 10, 171-192, (2012).
           http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
    .. [3] C. Walck, "Handbook on Statistical Distributions for
           Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
           University of Stockholm: Stockholm, Sweden, (2007).
           http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

    .. versionadded:: 1.1.0

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zmoyal_gen._shape_info  r   r1   Nc                 h    t                               dd||          }t          j        |           S )Nr   rH   )r~   r(   r   r   )r  r  rC   r   )r:   r   r   r  s       r/   r   zmoyal_gen._rvs  s3    YYAD$0  2 2r

{r1   c                     t          j        d|t          j        |           z   z            t          j        dt           j        z            z  S Nrq  rH   )rC   r   r   r   r   s     r/   re   zmoyal_gen._pdf  s:    vda"&!**n-..251A1AAAr1   c                 ~    t          j        t          j        d|z            t          j        d          z            S r  )rj   r  rC   r   r   r   s     r/   rh   zmoyal_gen._cdf  s-    wrvdQh''"'!**4555r1   c                 ~    t          j        t          j        d|z            t          j        d          z            S r  )rj   rr  rC   r   r   r   s     r/   rl   zmoyal_gen._sf  s-    vbfTAX&&3444r1   c                 \    t          j        dt          j        |          dz  z             S r	  )rC   r   rj   erfcinvr   s     r/   rq   zmoyal_gen._ppf  s'    q2:a==!++,,,,r1   c                     t          j        d          t           j        z   }t           j        dz  dz  }dt          j        d          z  t          j        d          z  t           j        dz  z  }d}||||fS )NrH      r  r  )rC   r   euler_gammar   r   rj   r  r4  s        r/   r   zmoyal_gen._stats  sa    VAYY'eQhl"'!**_rwqzz)BE1H43Br1   c                 b   |dk    r!t          j        d          t           j        z   S |dk    r7t           j        dz  dz  t          j        d          t           j        z   dz  z   S |dk    rwdt           j        dz  z  t          j        d          t           j        z   z  }t          j        d          t           j        z   dz  }dt	          j        d          z  }||z   |z   S |dk    rd	t	          j        d          z  t          j        d          t           j        z   z  }dt           j        dz  z  t          j        d          t           j        z   dz  z  }t          j        d          t           j        z   d
z  }dt           j        d
z  z  d
z  }||z   |z   |z   S |                     |          S )Nr|   rH   r   r  ri  r  r  r  8   r  rH  )rC   r   r  r   rj   r  r~  )r:   rU   tmp1r  tmp3tmp4s         r/   r   zmoyal_gen._munp  sc   886!99r~--#XX5!8a<26!99r~#="AAA#XX>RVAYYr~%=>DF1IIbn,q0D

?D$;%%#XXBGAJJ&"&))bn*DEDruax<26!99r~#="AADF1II.2Druax<!#D$;%,, ==###r1   r   )rv   rw   rx   ry   r^   r   re   rh   rl   rq   r   r   rz   r1   r/   r  r    s        ) )T     
B B B6 6 65 5 5- - -  $ $ $ $ $r1   r  moyalc                   R    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 ZddZddZd
S )nakagami_gena`  A Nakagami continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `nakagami` is:

    .. math::

        f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)

    for :math:`x >= 0`, :math:`\nu > 0`. The distribution was introduced in
    [2]_, see also [1]_ for further information.

    `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.

    %(after_notes)s

    References
    ----------
    .. [1] "Nakagami distribution", Wikipedia
           https://en.wikipedia.org/wiki/Nakagami_distribution
    .. [2] M. Nakagami, "The m-distribution - A general formula of intensity
           distribution of rapid fading", Statistical methods in radio wave
           propagation, Pergamon Press, 1960, 3-36.
           :doi:`10.1016/B978-0-08-009306-2.50005-4`

    %(example)s

    c                 @    t          dddt          j        fd          gS )NnuFr   r   r[   r]   s    r/   r^   znakagami_gen._shape_info  rP  r1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   r  s      r/   re   znakagami_gen._pdf  r  r1   c                     t          j        d          t          j        ||          z   t          j        |          z
  t          j        d|z  dz
  |          z   ||dz  z  z
  S r  )rC   r   rj   rf  rW  r  s      r/   r   znakagami_gen._logpdf  s^     q		BHR,,,rz"~~=21%%&(*1a40 	1r1   c                 8    t          j        |||z  |z            S rA   r[  r  s      r/   rh   znakagami_gen._cdf  s    {2r!tAv&&&r1   c                 \    t          j        d|z  t          j        ||          z            S r  rb  )r:   rp   r  s      r/   rq   znakagami_gen._ppf  s'    ws2vbnR333444r1   c                 8    t          j        |||z  |z            S rA   r^  r  s      r/   rl   znakagami_gen._sf  s    |B1Q'''r1   c                 \    t          j        d|z  t          j        ||          z            S rQ   rf  )r:   r}  r  s      r/   rt   znakagami_gen._isf   s'    wqtbob!444555r1   c                 P   t          j        |dz             t          j        |          z  t          j        |          z  }d||z  z
  }|dd|z  |z  z
  z  dz  |z  t          j        |d          z  }d|dz  z  |z  d|z  d	z
  |d	z  z  z   d	|z  z
  dz   }|||dz  z  z  }||||fS )
Nr   r|   r   r  r   ri  r  rH   )rj   r  rC   r   rj  )r:   r  r5  r6  r7  r8  s         r/   r   znakagami_gen._stats#  s    Xbfbhrll*272;;6"R%i1qtCx< 3&+bhsC.@.@@AXb[AbDFBE>)!B$.2
bck3Br1   Nc                 Z    t          j        |                    ||          |z            S rO  )rC   r   rg  )r:   r  r   r   s       r/   r   znakagami_gen._rvs+  s*    w|222D2AABFGGGr1   c                     |
d| j         z  }t          j        |          }t          j        t          j        ||z
  dz            t          |          z            }|||fz   S )N)r|   rH   )numargsrC   r  r   r  r  )r:   r   r;   r'   r(   s        r/   r  znakagami_gen._fitstart/  s]    <DL(D fTlls
Q//#d));<<sEl""r1   r   rA   )rv   rw   rx   ry   r^   re   r   rh   rq   rl   rt   r   r   r  rz   r1   r/   r  r    s         >F F F+ + +1 1 1' ' '5 5 5( ( (6 6 6  H H H H# # # # # #r1   r  nakagamic                 0   |dz  dz
  }t          j        |           t          j        |          }}t          j        |dz  | |z            d||z
  dz  z  z
  }t          j        |||z            dz  }t          |dk    ||fd t           j                   S )Nr   r|   r   rH   r   c                 0    | t          j        |          z   S rA   r#  )r  r  s     r/   r  z_ncx2_log_pdf.<locals>.<lambda>I  s    q26!99} r1   )r  r  )rC   r   rj   rf  iver   r\   )rd   rO  r  df2r  nsr  corrs           r/   _ncx2_log_pdfr  =  s     S&3,CWQZZB
(3s7AbD
!
!Cb1$4
4C6#r"u#Dq	d
$
$6'	   r1   c                   P    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 Zd Zd ZdS )ncx2_gena  A non-central chi-squared continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `ncx2` is:

    .. math::

        f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
            (x/\lambda)^{(k-2)/4}  I_{(k-2)/2}(\sqrt{\lambda x})

    for :math:`x >= 0`, :math:`k > 0` and :math:`\lambda \ge 0`.
    :math:`k` specifies the degrees of freedom (denoted ``df`` in the
    implementation) and :math:`\lambda` is the non-centrality parameter
    (denoted ``nc`` in the implementation). :math:`I_\nu` denotes the
    modified Bessel function of first order of degree :math:`\nu`
    (`scipy.special.iv`).

    `ncx2` takes ``df`` and ``nc`` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 F    |dk    t          j        |          z  |dk    z  S r:  r  r:   rO  r  s      r/   rV   zncx2_gen._argchecki  s"    Q"+b//)R1W55r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )NrO  Fr   r   r  rZ   r[   r:   idfincs      r/   r^   zncx2_gen._shape_infol  s>    uq"&k>BBuq"&k=AASzr1   Nc                 0    |                     |||          S rA   )noncentral_chisquare)r:   rO  r  r   r   s        r/   r   zncx2_gen._rvsq  s    00R>>>r1   c                 ~    t          j        |t                    |dk    z  }t          ||||ft          d           S )Nr
  r   c                 8    t                               | |          S rA   )rR  r   rd   rO  _s      r/   r  z"ncx2_gen._logpdf.<locals>.<lambda>w  s    dll1b.A.A r1   r  r  )rC   	ones_likeboolr   r  r:   rd   rO  r  conds        r/   r   zncx2_gen._logpdft  sL    |AT***bAg6$B}AAC C C 	Cr1   c                    t          j        |t                    |dk    z  }t          j                    5  d}t          j        d|           t          ||||ft          j        d           cd d d            S # 1 swxY w Y   d S )Nr
  r   z!overflow encountered in _ncx2_pdfr+  rn  c                 8    t                               | |          S rA   )rR  re   r  s      r/   r  zncx2_gen._pdf.<locals>.<lambda>      $))Ar2B2B r1   r  )	rC   r  r  rp  rq  rr  r   ra  	_ncx2_pdfr:   rd   rO  r  r  ro  s         r/   re   zncx2_gen._pdfy      |AT***bAg6$&& 	D 	D9G#Hg>>>>dQBK63C!B!BD D D	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D   9A<<B B c                     t          j        |t                    |dk    z  }t          ||||ft          j        d           S )Nr
  r   c                 8    t                               | |          S rA   )rR  rh   r  s      r/   r  zncx2_gen._cdf.<locals>.<lambda>  s    dii2.>.> r1   r  )rC   r  r  r   ra  	_ncx2_cdfr  s        r/   rh   zncx2_gen._cdf  sO    |AT***bAg6$Bv/?>>@ @ @ 	@r1   c                    t          j        |t                    |dk    z  }t          j                    5  d}t          j        d|           t          ||||ft          j        d           cd d d            S # 1 swxY w Y   d S )Nr
  r   z!overflow encountered in _ncx2_ppfr+  rn  c                 8    t                               | |          S rA   )rR  rq   r  s      r/   r  zncx2_gen._ppf.<locals>.<lambda>  r  r1   r  )	rC   r  r  rp  rq  rr  r   ra  	_ncx2_ppf)r:   rp   rO  r  r  ro  s         r/   rq   zncx2_gen._ppf  r  r  c                     t          j        |t                    |dk    z  }t          ||||ft          j        d           S )Nr
  r   c                 8    t                               | |          S rA   )rR  rl   r  s      r/   r  zncx2_gen._sf.<locals>.<lambda>  s    dhhq"oo r1   r  )rC   r  r  r   ra  _ncx2_sfr  s        r/   rl   zncx2_gen._sf  sJ    |AT***bAg6$Bv==? ? ? 	?r1   c                    t          j        |t                    |dk    z  }t          j                    5  d}t          j        d|           t          ||||ft          j        d           cd d d            S # 1 swxY w Y   d S )Nr
  r   z!overflow encountered in _ncx2_isfr+  rn  c                 8    t                               | |          S rA   )rR  rt   r  s      r/   r  zncx2_gen._isf.<locals>.<lambda>  r  r1   r  )	rC   r  r  rp  rq  rr  r   ra  	_ncx2_isfr  s         r/   rt   zncx2_gen._isf  r  r  c                     t          j        ||          t          j        ||          t          j        ||          t          j        ||          fS rA   )ra  
_ncx2_mean_ncx2_variance_ncx2_skewness_ncx2_kurtosis_excessr  s      r/   r   zncx2_gen._stats  sL    b"%%!"b))!"b))(R00	
 	
r1   r   )rv   rw   rx   ry   rV   r^   r   r   re   rh   rq   rl   rt   r   rz   r1   r/   r  r  M  s         66 6 6  
? ? ? ?C C C
D D D@ @ @
D D D? ? ?
D D D
 
 
 
 
r1   r  ncx2c                   R    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 Zd ZddZdS )ncf_gena  A non-central F distribution continuous random variable.

    %(before_notes)s

    See Also
    --------
    scipy.stats.f : Fisher distribution

    Notes
    -----
    The probability density function for `ncf` is:

    .. math::

        f(x, n_1, n_2, \lambda) =
            \exp\left(\frac{\lambda}{2} +
                      \lambda n_1 \frac{x}{2(n_1 x + n_2)}
                \right)
            n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
            (n_2 + n_1 x)^{-(n_1 + n_2)/2}
            \gamma(n_1/2) \gamma(1 + n_2/2) \\
            \frac{L^{\frac{n_1}{2}-1}_{n_2/2}
                \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
            {B(n_1/2, n_2/2)
                \gamma\left(\frac{n_1 + n_2}{2}\right)}

    for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`.  Here :math:`n_1` is the
    degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
    the denominator, :math:`\lambda` the non-centrality parameter,
    :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
    generalized Laguerre polynomial and :math:`B` is the beta function.

    `ncf` takes ``df1``, ``df2`` and ``nc`` as shape parameters. If ``nc=0``,
    the distribution becomes equivalent to the Fisher distribution.

    %(after_notes)s

    %(example)s

    c                 *    |dk    |dk    z  |dk    z  S r:  rz   )r:   df1r  r  s       r/   rV   zncf_gen._argcheck  s    aC!G$a00r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }t          dddt          j        fd          }|||gS )Nr  Fr   r   r  r  rZ   r[   )r:   idf1idf2r  s       r/   r^   zncf_gen._shape_info  sZ    %BF^DD%BF^DDuq"&k=AAdC  r1   Nc                 2    |                     ||||          S rA   )noncentral_f)r:   rK  rL  r  r   r   s         r/   r   zncf_gen._rvs  s    ((c2t<<<r1   c                 0    t          j        ||||          S rA   )ra  _ncf_pdfr:   rd   rK  rL  r  s        r/   re   zncf_gen._pdf  s     q#sB///r1   c                 0    t          j        ||||          S rA   )ra  _ncf_cdfr  s        r/   rh   zncf_gen._cdf      q#sB///r1   c                 0    t          j        ||||          S rA   )ra  _ncf_ppf)r:   rp   rK  rL  r  s        r/   rq   zncf_gen._ppf  r  r1   c                 0    t          j        ||||          S rA   )ra  _ncf_sfr  s        r/   rl   zncf_gen._sf  s    ~ac2...r1   c                 0    t          j        ||||          S rA   )ra  _ncf_isfr  s        r/   rt   zncf_gen._isf  r  r1   c                 <   |dz  |z  |z  }t          j        |d|z  z             t          j        d|z  |z
            z   t          j        |dz            z
  }|t          j        | dz  |z             z  }|t          j        |d|z  z   d|z  d|z            z  }|S )Nr|   r   r   )rj   rW  rC   r   hyp1f1)r:   rU   rK  rL  r  r  terms          r/   r   zncf_gen._munp  s    Sy}q z!CG)$$rz#c'!)'<'<<rz#c'?R?RRrvrcCin%%%ry3s7CGSV444
r1   r  c                     t          j        |||          }t          j        |||          }d|v rt          j        |||          nd }d|v rt          j        |||          nd }||||fS )Nr  r  )ra  	_ncf_mean_ncf_variance_ncf_skewness_ncf_kurtosis_excess)	r:   rK  rL  r  r  r5  r6  r7  r8  s	            r/   r   zncf_gen._stats  s    c3++"3R0036'>>V!#sB///t G^^ (b  15 	3Br1   r   r  )rv   rw   rx   ry   rV   r^   r   re   rh   rq   rl   rt   r   r   rz   r1   r/   r  r    s        ' 'P1 1 1! ! != = = =0 0 00 0 00 0 0/ / /0 0 0       r1   r  ncfc                   P    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
d
 Zd Zd ZdS )t_gena  A Student's t continuous random variable.

    For the noncentral t distribution, see `nct`.

    %(before_notes)s

    See Also
    --------
    nct

    Notes
    -----
    The probability density function for `t` is:

    .. math::

        f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
                        {\sqrt{\pi \nu} \Gamma(\nu/2)}
                    (1+x^2/\nu)^{-(\nu+1)/2}

    where :math:`x` is a real number and the degrees of freedom parameter
    :math:`\nu` (denoted ``df`` in the implementation) satisfies
    :math:`\nu > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS rN  r[   r]   s    r/   r^   zt_gen._shape_info#  rP  r1   Nc                 0    |                     ||          S rO  )
standard_trS  s       r/   r   z
t_gen._rvs&  s    &&r&555r1   c                 L    t          |t          j        k    ||fd d           S )Nc                 6    t                               |           S rA   )r   re   rd   rO  s     r/   r  zt_gen._pdf.<locals>.<lambda>,  s    DIIaLL r1   c                     t          j        t          j        |dz   dz            t          j        |dz            z
            t          j        |t           j        z            d| dz  |z  z   |dz   dz  z  z  z  S r"  )rC   r   rj   rW  r   r   r  s     r/   r  zt_gen._pdf.<locals>.<lambda>-  sp    rz2a4(++BJr!t,<,<<==72be8$$aAr	kbdAX%>>@ r1   r  r  rU  s      r/   re   z
t_gen._pdf)  s8    "&L1b'(( 
 
 
 	
r1   c                 L    t          |t          j        k    ||fd d           S )Nc                 6    t                               |           S rA   )r   r   r  s     r/   r  zt_gen._logpdf.<locals>.<lambda>6  s    DLLOO r1   c                     t          j        |dz   dz            t          j        |dz            z
  dt          j        |t          j        z            z  |dz   dz  t          j        d| dz  |z  z             z  z   z
  S r  )rj   rW  rC   r   r   r  s     r/   r  zt_gen._logpdf.<locals>.<lambda>7  sw    
BqD!8$$rz"Q$'7'77rvbh'''dAXbfQ1by[11123 r1   r  r  rU  s      r/   r   zt_gen._logpdf3  s8    "&L1b'++ 
 
 
 	
r1   c                 ,    t          j        ||          S rA   rj   stdtrrU  s      r/   rh   z
t_gen._cdf>  rv  r1   c                 .    t          j        ||           S rA   r  rU  s      r/   rl   z	t_gen._sfA  s    xQBr1   c                 ,    t          j        ||          S rA   rj   stdtritrd  s      r/   rq   z
t_gen._ppfD  s    z"a   r1   c                 .    t          j        ||           S rA   r  rd  s      r/   rt   z
t_gen._isfG  s    
2q!!!!r1   c                    t          j        |          }t          j        |dk    dt           j                  }|dk    |dk    z  |dk    t          j        |          z  |f}d d d f}t          |||ft           j                  }t          j        |dk    dt           j                  }|dk    |dk    z  |dk    t          j        |          z  |f}d	 d
 d f}t          |||ft           j                  }||||fS )Nr   r{   rH   c                 J    t          j        t           j        | j                  S rA   rC   broadcast_tor\   r  rO  s    r/   r  zt_gen._stats.<locals>.<lambda>S      !B!B r1   c                     | | dz
  z  S r0  rz   r  s    r/   r  zt_gen._stats.<locals>.<lambda>T  s    r#v r1   c                 6    t          j        d| j                  S rQ   rC   r  r  r  s    r/   r  zt_gen._stats.<locals>.<lambda>U      BH!=!= r1   r  r  c                 J    t          j        t           j        | j                  S rA   r  r  s    r/   r  zt_gen._stats.<locals>.<lambda>]  r  r1   c                     d| dz
  z  S )Nr  r  rz   r  s    r/   r  zt_gen._stats.<locals>.<lambda>^  s    3 r1   c                 6    t          j        d| j                  S r:  r  r  s    r/   r  zt_gen._stats.<locals>.<lambda>_  r  r1   )rC   isposinfr  r\   r   r   r  )	r:   rO  infinite_dfr5  rZ  
choicelistr6  r7  r8  s	            r/   r   zt_gen._statsJ  s   k"ooXb1fc26**!Va(!Vr{2.! CB..==?
 (Jrv>>Xb1fc26**!Va(!Vr{2.! CB//==?
 :ubf==3Br1   c                 @   |t           j        k    rt                                          S |dz  }|dz   dz  }|t	          j        |          t	          j        |          z
  z  t          j        t          j        |          t	          j        |d          z            z   S )NrH   r   r   )	rC   r\   r   r   rj   rT  r   r   r_  )r:   rO  halfhalf1s       r/   r   zt_gen._entropyd  s    <<==??"!ta
rz%((2:d+;+;;<&RWT3%7%77889 	:r1   r   )rv   rw   rx   ry   r^   r   re   r   rh   rl   rq   rt   r   r   rz   r1   r/   r   r     s         <F F F6 6 6 6
 
 
	
 	
 	
       ! ! !" " "  4: : : : :r1   r   r$  c                   L    e Zd ZdZd Zd ZddZd Zd Zd Z	d	 Z
d
 ZddZdS )nct_gena  A non-central Student's t continuous random variable.

    %(before_notes)s

    Notes
    -----
    If :math:`Y` is a standard normal random variable and :math:`V` is
    an independent chi-square random variable (`chi2`) with :math:`k` degrees
    of freedom, then

    .. math::

        X = \frac{Y + c}{\sqrt{V/k}}

    has a non-central Student's t distribution on the real line.
    The degrees of freedom parameter :math:`k` (denoted ``df`` in the
    implementation) satisfies :math:`k > 0` and the noncentrality parameter
    :math:`c` (denoted ``nc`` in the implementation) is a real number.

    %(after_notes)s

    %(example)s

    c                     |dk    ||k    z  S r:  rz   r  s      r/   rV   znct_gen._argcheck  s    Q28$$r1   c                     t          dddt          j        fd          }t          ddt          j         t          j        fd          }||gS )NrO  Fr   r   r  r[   r  s      r/   r^   znct_gen._shape_info  sC    uq"&k>BBuw&7HHSzr1   Nc                     t                               |||          }t                              |||          }|t          j        |          z  t          j        |          z  S )Nr@  r  )r   r  rR  rC   r   )r:   rO  r  r   r   rU   r4  s          r/   r   znct_gen._rvs  sO    HH$\HBBXXbt,X??272;;,,r1   c                 D   |dz  }|dz  }||z  }||z  |z  }||z   }|dz  t          j        |          z  t          j        |dz             z   |t          j        d          z  ||z  dz  z   |dz  t          j        |          z  z   t          j        |dz            z   z
  }t          j        |          }	|d|z  z  }
t          j        d          |z  |z  t          j        |dz  dz   d|
          z  t          j        |t          j        |dz   dz            z            z  }t          j        |dz   dz  d|
          t          j        t          j        |          t          j        |dz  dz             z            z  }|	||z   z  }	t          j	        |	dd           S )Nr|   r   r   rH   ri  r   r   )
rC   r   rj   rW  r   r   r  r   r  clip)r:   rd   rO  r  rU   r#  r  r  trm1r  valFtrm2s               r/   re   znct_gen._pdf  s   sFVqS"uRx2v"RVAYYAaC0RVAYY;Bq(AaC+==Z!__%& VD\\qv

2a	!A#a%d ; ;;*T"(AaC7"3"33445	1Q3'3--*RWT]]28AaCE??:;;<
d4iwr1d###r1   c                 V    t          j        t          j        |||          dd          S Nr   r   )rC   r,  ra  _nct_cdfr:   rd   rO  r  s       r/   rh   znct_gen._cdf  s$    wvq"b111a888r1   c                 .    t          j        |||          S rA   )ra  _nct_ppf)r:   rp   rO  r  s       r/   rq   znct_gen._ppf      q"b)))r1   c                 V    t          j        t          j        |||          dd          S r1  )rC   r,  ra  _nct_sfr3  s       r/   rl   znct_gen._sf  s$    wv~aR00!Q777r1   c                 .    t          j        |||          S rA   )ra  _nct_isfr3  s       r/   rt   znct_gen._isf  r6  r1   r  c                     t          j        ||          }t          j        ||          }d|v rt          j        ||          nd }d|v rt          j        ||          dz
  nd }||||fS )Nr  r  r  )ra  	_nct_mean_nct_variance_nct_skewness_nct_kurtosis_excess)r:   rO  r  r  r5  r6  r7  r8  s           r/   r   znct_gen._stats  sw    b"%%"2r**-0G^^V!"b)))69WnnV(R0022$3Br1   r   r  )rv   rw   rx   ry   rV   r^   r   re   rh   rq   rl   rt   r   rz   r1   r/   r'  r'  p  s         0% % %  
- - - -
$ $ $&9 9 9* * *8 8 8* * *     r1   r'  nctc                        e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
e ee           fd
                        Z xZS )
pareto_genaL  A Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pareto` is:

    .. math::

        f(x, b) = \frac{b}{x^{b+1}}

    for :math:`x \ge 1`, :math:`b > 0`.

    `pareto` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zpareto_gen._shape_info  r   r1   c                     ||| dz
  z  z  S rQ   rz   r  s      r/   re   zpareto_gen._pdf  s    1r!t9}r1   c                     d|| z  z
  S rQ   rz   r  s      r/   rh   zpareto_gen._cdf  s    1r7{r1   c                 .    t          d|z
  d|z            S )Nr   r  r  r  s      r/   rq   zpareto_gen._ppf  s    1Q3Qr1   c                     || z  S rA   rz   r  s      r/   rl   zpareto_gen._sf  s    A2wr1   r  c                 X   d\  }}}}d|v ri|dk    }t          j        ||          }t          j        t          j        |          t           j                  }t          j        ||||dz
  z             d|v rr|dk    }t          j        ||          }t          j        t          j        |          t           j                  }t          j        ||||dz
  z  |dz
  dz  z             d	|v r|d
k    }t          j        ||          }t          j        t          j        |          t           j                  }d|dz   z  t          j        |dz
            z  |dz
  t          j        |          z  z  }	t          j        |||	           d|v r|dk    }t          j        ||          }t          j        t          j        |          t           j                  }dt          j        g d|          z  t          j        g d|          z  }	t          j        |||	           ||||fS )NNNNNrT  r   rE  r|   r  rH   r   r  r  r  r  r  r  )r|   r|   r  r  )r|   g      r  r{   )	rC   extractrG  r  r\   placer  r   r  )
r:   r   r  r5  r6  r7  r8  maskbtr  s
             r/   r   zpareto_gen._stats  s   0CR'>>q5DD!$$B!888BHRrRV}---'>>q5DD!$$B'"(1++"&999CHS$bfC! ;<<<'>>q5DD!$$B!888BS>BGBH$5$55"s(bgbkk9QRDHRt$$$'>>q5DD!$$B!888B
#5#5#5r:::J555r::;DHRt$$$3Br1   c                 <    dd|z  z   t          j        |          z
  S r  r#  r)  s     r/   r   zpareto_gen._entropy      3q5y26!99$$r1   c                    t          | ||          }|\  }}|9t          j                  |z
  |pdk     rt          ddt          j                  j        d         fd||cxu r6n n2fdfdfdfd	}|                    d
d          }|dz  |dz  }
}	 ||	|
          sB|	dk    s|
t          j        k     r,|	dz  }	|
dz  }
 ||	|
          s|	dk    |
t          j        k     ,t          |	|
g          }|j        rx|j	        }t          j                  |z
  }p ||          }||z   t          j                  k     s,t          j                  |z
  }t          j
        |d          }|||fS  t                      j        fi |S |t          j                  |z
  }n|}|pt          j                  |z
  }p ||          }|||fS )Nr   paretor   r  c                 b    t          j        t          j        |z
  | z                      z  S rA   )rC   r  r   )r(   locationr   ndatas     r/   	get_shapez!pareto_gen.fit.<locals>.get_shape  s-     26"&$/U)B"C"CDDDDr1   c                     | z  |z  S rA   rz   )r  r(   rT  s     r/   	dL_dScalez!pareto_gen.fit.<locals>.dL_dScale  s     u}u,,r1   c                 D    | dz   t          j        d|z
  z            z  S rQ   rC   r  )r  rS  r   s     r/   dL_dLocationz$pareto_gen.fit.<locals>.dL_dLocation!  s'     	RVA,A%B%BBBr1   c                     t          j                  | z
  }p | |          } ||           ||           z
  S rA   )rC   r  )r(   rS  r  rZ  rW  r   fshaperU  s      r/   fun_to_solvez$pareto_gen.fit.<locals>.fun_to_solve&  sP     6$<<%/<))E8"<"<#|E844yy7N7NNNr1   c                 |    t          j         |                     t          j         |                    k    S rA   rB   rE   rF   r]  s     r/   rG   z.pareto_gen.fit.<locals>.interval_contains_root-  s;    V 4 455V 4 4556 7r1   r(   rH   r  )r  rC   r  r<  r\   r  r5   r$   	convergedr  	nextafterr7   r9   )r:   r   r;   r.   
parametersr   r   rG   r  rE   rF   r  r(   r'   r  rZ  rW  r\  r]  rU  rT  r  s    `             @@@@@@r/   r9   zpareto_gen.fit  s    1tT4HH
%/"fdF tt 3v{ C Cxq????
1	E 	E 	E 	E 	E 	E
 6!!!!!!!!- - - - -
C C C C C
O O O O O O O O O7 7 7 7 7 ((7A..K(1_kAoFF .-ff== 

frvoo!! .-ff== 

frvoo lVV4DEEEC} 1fTllU*7))E3"7"7 rvd||33F4LL3.EL22Ec5(("uww{4004000\&,,'CCC ,"&,,,/))E3//c5  r1   r  )rv   rw   rx   ry   r^   re   rh   rq   rl   r   r   r>   r
   r   r9   r  r  s   @r/   rB  rB    s         *E E E              6% % % M**R! R! R! R! +* _R! R! R! R! R!r1   rB  rQ  c                   H    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZdS )	lomax_gena  A Lomax (Pareto of the second kind) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lomax` is:

    .. math::

        f(x, c) = \frac{c}{(1+x)^{c+1}}

    for :math:`x \ge 0`, :math:`c > 0`.

    `lomax` takes ``c`` as a shape parameter for :math:`c`.

    `lomax` is a special case of `pareto` with ``loc=-1.0``.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zlomax_gen._shape_infov  r   r1   c                 $    |dz  d|z   |dz   z  z  S r  rz   r  s      r/   re   zlomax_gen._pdfy  s    uc!equ%%%r1   c                 `    t          j        |          |dz   t          j        |          z  z
  S rQ   r'  r  s      r/   r   zlomax_gen._logpdf}  s&    vayyAaC!,,,r1   c                 X    t          j        | t          j        |          z             S rA   r  r  s      r/   rh   zlomax_gen._cdf  s#    !BHQKK((((r1   c                 V    t          j        | t          j        |          z            S rA   )rC   r   rj   r  r  s      r/   rl   zlomax_gen._sf  s     vqb!n%%%r1   c                 2    | t          j        |          z  S rA   r  r  s      r/   r   zlomax_gen._logsf  s    r"(1++~r1   c                 X    t          j        t          j        |            |z            S rA   r  r  s      r/   rq   zlomax_gen._ppf  s"    x1"a(((r1   c                 R    t                               |dd          \  }}}}||||fS )Nr  rz  )r'   r  )rQ  r  r  s         r/   r   zlomax_gen._stats  s/     ,,qdF,CCCR3Br1   c                 <    dd|z  z   t          j        |          z
  S r  r#  r)  s     r/   r   zlomax_gen._entropy  s    Qwrvayy  r1   N)rv   rw   rx   ry   r^   re   r   rh   rl   r   rq   r   r   rz   r1   r/   re  re  ^  s         .E E E& & &- - -) ) )& & &  ) ) )  ! ! ! ! !r1   re  lomaxc                        e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
dd
Zd Ze eed           fd                        Z xZS )pearson3_gena  A pearson type III continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pearson3` is:

    .. math::

        f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
                       (\beta (x - \zeta))^{\alpha - 1}
                       \exp(-\beta (x - \zeta))

    where:

    .. math::

            \beta = \frac{2}{\kappa}

            \alpha = \beta^2 = \frac{4}{\kappa^2}

            \zeta = -\frac{\alpha}{\beta} = -\beta

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).
    Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
    ``skew``.

    %(after_notes)s

    %(example)s

    References
    ----------
    R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
    Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
    Resources Research, Vol.27, 3149-3158 (1991).

    L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
    Vol.1, 191-198 (1930).

    "Using Modern Computing Tools to Fit the Pearson Type III Distribution to
    Aviation Loads Data", Office of Aviation Research (2003).

    c                    d}d}d}t          j        d||          \  }}}|                                }t          j        |          |k     }| }d||         |z  z  }	||	z  dz  }
||
|	z  z
  }|	||         |z
  z  }||||||	|
|fS )Nr{   r|   g>r   rH   )rC   r  copyr  )r:   rd   r  r'   r(   norm2pearson_transitionansrL  invmaskr_  r  r  transxs                r/   _preprocesszpearson3_gen._preprocess  s    
  #+*3488Qhhjj {4  #::%d7me+,!UT\!7d*+AvtWdE4??r1   c                 *    t          j        |          S rA   r  )r:   r  s     r/   rV   zpearson3_gen._argcheck  s    
 {4   r1   c                 V    t          ddt          j         t          j        fd          gS )Nr  Fr   r[   r]   s    r/   r^   zpearson3_gen._shape_info  s$    65BF7BF*;^LLMMr1   c                 *    d}d}|}d|dz  z  }||||fS )Nr{   r|   ri  rH   rz   )r:   r  rT  r  r  r  s         r/   r   zpearson3_gen._stats  s,    aK!Qzr1   c                     t          j        |                     ||                    }|j        dk    rt          j        |          rdS |S d|t          j        |          <   |S )Nr   r{   )rC   r   r   r  r  )r:   rd   r  ru  s       r/   re   zpearson3_gen._pdf  s]    
 fT\\!T**++8q==x}} sJ BHSMM
r1   c                    |                      ||          \  }}}}}}}}	t          j        t          ||                             ||<   t          j        t	          |                    t
                              ||          z   ||<   |S rA   )rx  rC   r   r   r  r  r-  )
r:   rd   r  ru  rw  rL  rv  r_  r  r  s
             r/   r   zpearson3_gen._logpdf  s     Q%% 	6QgtUA F9QtW--..D	 vc$ii((5<<+F+FFG
r1   c                    |                      ||          \  }}}}}}}}t          ||                   ||<   t          j        ||j                  }t          j        ||dk              }	||         dk    }
t                              ||
         ||
                   ||	<   t          j        ||dk               }||         dk     }t                              ||         ||                   ||<   |S r:  )	rx  r   rC   r  r  r  r  r   r!  )r:   rd   r  ru  rw  rL  rv  r  r  	invmask1a	invmask1b	invmask2a	invmask2bs                r/   rh   zpearson3_gen._cdf  s    Q%% 	3Qgq% ag&&D	tW]33N7D1H55	MA%	 6)#4eI6FGGI N7D1H55	MA%	&"3U95EFFI
r1   Nc                 6   t          j        ||          }|                     dg|          \  }}}}}}}	}
|                                }|j        |z
  }|                    |          ||<   |                    |	|          |z  |
z   ||<   |dk    r|d         }|S )Nr   rz   )rC   r  rx  r  r   r   rg  )r:   r  r   r   ru  r  rL  rv  r_  r  r  nsmallnbigs                r/   r   zpearson3_gen._rvs&  s    tT**aS$'' 	4Q4$t y6! 0088D	#225$??DtKG2::a&C
r1   c                     |                      ||          \  }}}}}}}}	t          ||                   ||<   ||         }d||dk              z
  ||dk     <   t          j        ||          |z  |	z   ||<   |S r  )rx  r   rj   rc  )
r:   rp   r  ru  r  rL  rv  r_  r  r  s
             r/   rq   zpearson3_gen._ppf4  s    Q%% 	4Q4$tag&&D	gJ!D1H+o$(~eQ//4t;G
r1   ze        Note that method of moments (`method='MM'`) is not
        available for this distribution.

r   c                     |                     dd           dk    rt          d           t          t          |           |           j        |g|R i |S )Nr*   MMzhFit `method='MM'` is not available for the Pearson3 distribution. Please try the default `method='MLE'`.)r5   r  r7   r8   r9   r  s       r/   r9   zpearson3_gen.fit=  sn    
 88Hd##t++% 'D E E E /5dT**.tCdCCCdCCCr1   r   )rv   rw   rx   ry   rx  rV   r^   r   re   r   rh   r   rq   r>   r   r   r9   r  r  s   @r/   rq  rq    s       , ,Z@ @ @8! ! !N N N        0      } 50 1 1 1D D D D1 1 _D D D D Dr1   rq  pearson3c                        e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Z fd
Ze eed           fd                        Z xZS )powerlaw_gena  A power-function continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto

    Notes
    -----
    The probability density function for `powerlaw` is:

    .. math::

        f(x, a) = a x^{a-1}

    for :math:`0 \le x \le 1`, :math:`a > 0`.

    `powerlaw` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    For example, the support of `powerlaw` can be adjusted from the default
    interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and
    ``scale=d``. For a power-law distribution with infinite support, see
    `pareto`.

    `powerlaw` is a special case of `beta` with ``b=1``.

    %(example)s

    c                 @    t          dddt          j        fd          gS r   r[   r]   s    r/   r^   zpowerlaw_gen._shape_infon  r   r1   c                     |||dz
  z  z  S r  rz   r  s      r/   re   zpowerlaw_gen._pdfq  s    QsU|r1   c                 \    t          j        |          t          j        |dz
  |          z   S rQ   )rC   r   rj   rf  r  s      r/   r   zpowerlaw_gen._logpdfu  s%    vayy28AE1----r1   c                     ||dz  z  S r  rz   r  s      r/   rh   zpowerlaw_gen._cdfx  s    1S5zr1   c                 0    |t          j        |          z  S rA   r#  r  s      r/   r   zpowerlaw_gen._logcdf{  r$  r1   c                 (    t          |d|z            S r  r  r  s      r/   rq   zpowerlaw_gen._ppf~  s    1c!e}}r1   c                     ||dz   z  ||dz   z  |dz   dz  z  d|dz
  |dz   z  z  t          j        |dz   |z            z  dt          j        g d|          z  ||dz   z  |dz   z  z  fS )	Nr|   r   rH   g       r  r  )r   r  r  rH   r  )rC   r   r  r  s     r/   r   zpowerlaw_gen._stats  s    QWQWSQ.SQW-.!c'Q1G1GGBJ~~~q111Q!c']a!e5LMO 	Or1   c                 <    dd|z  z
  t          j        |          z
  S r  r#  r  s     r/   r   zpowerlaw_gen._entropy  rO  r1   c                 r    t          t          |                               ||          |dk    |dk    z  z  S r1  )r7   r  r  )r:   rd   r~   r  s      r/   r  zpowerlaw_gen._support_mask  s:    lD))771==FqAv&( 	)r1   a:          Notes specifically for ``powerlaw.fit``: If the location is a free
        parameter and the value returned for the shape parameter is less than
        one, the true maximum likelihood approaches infinity. This causes
        numerical difficulties, and the resulting estimates are approximate.
        

r   c                 N   |                     dd          r t                      j        g|R i |S t          t	          j                            dk    r t                      j        g|R i |S t          | ||          \  }}|                               fg}|                     |i           d         }|W	                                |k    st          ddd          |,                                ||z   k    st          ddd          |>|dk    rt          d          |                                k    rd}t          |          d d	 || ||          ||fS |t	          j        	                                t          j                   }	p |	|          }
 ||
|	|f          }t	          j                                        |z
  t          j                  }p ||          } ||||f          }||k     r|
|	|fS |||fS |  |          }p ||          }|||fS fd
}d d fdfdfd}dk    r
 |            S dk    r
 |            S  |            }|                     |          } |            }|                     |          }||k    r|d         dk    r|S ||k    r|d         dk    r|S  t                      j        g|R i |S )Nr  Fr   powerlawr   zKNegative or zero `fscale` is outside the range allowed by the distribution.z0`fscale` must be greater than the range of data.c                     t          |           }| t          j        t          j        | |z
                      |t          j        |          z  z
  z  S rA   )r  rC   r  r   )r   r'   r(   r  s       r/   rU  z#powerlaw_gen.fit.<locals>.get_shape  sE     D		A3"&s
!3!344qFGGr1   c                 0    |                                  |z
  S rA   )r  )r   r'   s     r/   	get_scalez#powerlaw_gen.fit.<locals>.get_scale  s     88::##r1   c                     t          j                                        t           j                   } t          j        |           t          j        | j                  j        k     r3t          j        |           t          j        | j                  j        z  } t          j         |           t           j                  }p | |          }|| |fS rA   )	rC   rb  r  r\   r  finfor  tinyrD   )r'   r(   r  r   r\  r  rU  s      r/   fit_loc_scale_w_shape_lt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_lt_1   s    ,txxzzBF733Cvc{{RXci00555gcllRXci%8%8%==L4!5!5rv>>E9iic599E#u$$r1   c                 *    | j         d          |z  |z  S r:  )r  )r   r  r(   s      r/   rW  z#powerlaw_gen.fit.<locals>.dL_dScale  s     JqM>E)E11r1   c                 B    |dz
  t          j        d|| z
  z            z  S rQ   rY  )r   r  r'   s      r/   rZ  z&powerlaw_gen.fit.<locals>.dL_dLocation  s&     AIS4Z(8!9!999r1   c                     t          j         |           t           j                   }p | |          } ||           S rA   rC   rb  r\   )r'   r(   r  rZ  r   r\  r  rU  s      r/   dL_dLocation_starz+powerlaw_gen.fit.<locals>.dL_dLocation_star  sR     L4!5!5w??E9iic599E<eS111r1   c                     t          j         |           t           j                   }p | |          } ||           ||           z
  S rA   r  )	r'   r(   r  rZ  rW  r   r\  r  rU  s	      r/   r]  z&powerlaw_gen.fit.<locals>.fun_to_solve   sj     L4!5!5w??E9iic599EIdE511"l4445 6r1   c                     t          j        
                                t           j                   } 
                                | z
  } 	|           dk    r+
                                |z
  } |dz  } 	|           dk    +fd}| dz
  }d} |||           sJ|t           j         k    r9
                                |z
  }|dz  } |||           s|t           j         k    9t	          j        || f          }t          j        |j        t           j                   }t          j         
|          t           j                  }p 
||          }|||fS )Nr   rH   c                 |    t          j         |                     t          j         |                    k    S rA   rB   r_  s     r/   rG   zTpowerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1.<locals>.interval_contains_root4  s;    V 4 4557<<#7#7889 :r1   r   r|   r`  )rC   rb  r  r\   r   r$   r  )rF   r(  rG   rE   r  r  r'   r(   r  r  r   r\  r]  r  rU  s            r/   fit_loc_scale_w_shape_gt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1(  s    \$((**rvg66F XXZZ&(E##F++a//e+
 $#F++a//: : : : :
 aZF
 A--ff== "&((((**q.Q .-ff== "&(( 'vv>NOOOD,ty26'22CL4!5!5rv>>E9iic599E#u$$r1   )r,   r7   r9   r  rC   uniquer  r  _reduce_funcr  r<  r  r   ptprb  r\   nnlf)r:   r   r;   r.   r   r   penalized_nllf_argspenalized_nllfrL   loc_lt1	shape_lt1ll_lt1loc_gt1	shape_gt1ll_gt1r(   r  r  r  fit_shape_lt1fit_shape_gt1rZ  r  rW  r\  r]  r  rU  r  s    `                   @@@@@@@r/   r9   zpowerlaw_gen.fit  s   P 88J&& 	4577;t3d333d333ry1$$577;t3d333d333%@tAEt&M &M"fdF#dnnT&:&:%<=**+>CCAF
 88::$$":q!444!$((**v*E*E":q!444{{  "F G G G##H oo%	H 	H 	H	$ 	$ 	$ $"29T400$>> l488::w77GB))D'6"B"BI#^Y$@$GGF l488::#6??GB))D'6"B"BI#^Y$@$GGF '611 '611 IdD))E:iidE::E$%%
	% 	% 	% 	% 	% 	% 	% 	%	2 	2 	2
	: 	: 	:
	2 	2 	2 	2 	2 	2 	2 	2 	2	6 	6 	6 	6 	6 	6 	6 	6 	6 	6!	% !	% !	% !	% !	% !	% !	% !	% !	% !	%H &A++--///FQJJ--/// 3244=$//2244=$//Va 0A 5 5  f__q!1A!5!5  577;t3d333d333r1   )rv   rw   rx   ry   r^   re   r   rh   r   rq   r   r   r  r>   r   r   r9   r  r  s   @r/   r  r  M  s        @E E E  . . .      O O O% % %) ) ) ) ) } 5   H4 H4 H4 H4  _H4 H4 H4 H4 H4r1   r  r  c                   8    e Zd ZdZej        Zd Zd Zd Z	d Z
dS )powerlognorm_gena  A power log-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powerlognorm` is:

    .. math::

        f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
                     (\Phi(-\log(x)/s))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x > 0`, :math:`s, c > 0`.

    `powerlognorm` takes :math:`c` and :math:`s` as shape parameters.

    %(after_notes)s

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )Nr  Fr   r   r  r[   )r:   r  r6  s      r/   r^   zpowerlognorm_gen._shape_info}  r7  r1   c                     |||z  z  t          t          j        |          |z            z  t          t	          t          j        |           |z            |dz  dz
            z  S r  )r   rC   r   r  r   r:   rd   r  r  s       r/   re   zpowerlognorm_gen._pdf  s]     1Q3)BF1IIaK000Irvayyjl++QsU3Y778 	9r1   c                 t    dt          t          t          j        |           |z            |dz            z
  S r  )r  r   rC   r   r  s       r/   rh   zpowerlognorm_gen._cdf  s1    SBF1II:a<00!C%8888r1   c           
      t    t          j        | t          t          d|z
  d|z                      z            S r  )rC   r   r   r  )r:   rp   r  r  s       r/   rq   zpowerlognorm_gen._ppf  s3    vqb9Sq#'%:%:;;;<<<r1   N)rv   rw   rx   ry   r   r  r  r^   re   rh   rq   rz   r1   r/   r  r  c  sd         . "4M  
9 9 99 9 9= = = = =r1   r  powerlognormc                   0    e Zd ZdZd Zd Zd Zd Zd ZdS )powernorm_gena  A power normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powernorm` is:

    .. math::

        f(x, c) = c \phi(x) (\Phi(-x))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x >= 0`, :math:`c > 0`.

    `powernorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zpowernorm_gen._shape_info  r   r1   c                 T    |t          |          z  t          |           |dz
  z  z  S r  r   r   r  s      r/   re   zpowernorm_gen._pdf  s(    1~A23!788r1   c                 x    t          j        |          t          |          z   |dz
  t          |           z  z   S rQ   )rC   r   r   r   r  s      r/   r   zpowernorm_gen._logpdf  s3    vayy<??*ac<3C3C-CCCr1   c                 4    dt          |           |dz  z  z
  S r  r   r  s      r/   rh   zpowernorm_gen._cdf  s    9aR==1S5)))r1   c                 J    t          t          d|z
  d|z                       S r  )r   r  r  s      r/   rq   zpowernorm_gen._ppf  s%    #cAgsQw//0000r1   Nr  rz   r1   r/   r  r    so         ,E E E9 9 9D D D* * *1 1 1 1 1r1   r  	powernormc                   >    e Zd ZdZd Zd Zd Zd Zd Zd
dZ	d	 Z
dS )	rdist_gena/  An R-distributed (symmetric beta) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rdist` is:

    .. math::

        f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}

    for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
    symmetric beta distribution: if B has a `beta` distribution with
    parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
    parameter c.

    `rdist` takes ``c`` as a shape parameter for :math:`c`.

    This distribution includes the following distribution kernels as
    special cases::

        c = 2:  uniform
        c = 3:  `semicircular`
        c = 4:  Epanechnikov (parabolic)
        c = 6:  quartic (biweight)
        c = 8:  triweight

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zrdist_gen._shape_info  r   r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zrdist_gen._pdf  r  r1   c                 ~    t          j        d           t                              |dz   dz  |dz  |dz            z   S r  )rC   r   r_  r   r  s      r/   r   zrdist_gen._logpdf  s7    q		zDLL!a%AaC1====r1   c                 R    t                               |dz   dz  |dz  |dz            S r"  )r_  rh   r  s      r/   rh   zrdist_gen._cdf  s(    yy!a%AaC1---r1   c                 R    dt                               ||dz  |dz            z  dz
  S r  )r_  rq   r  s      r/   rq   zrdist_gen._ppf  s*    1ac1Q3'''!++r1   Nc                 H    d|                     |dz  |dz  |          z  dz
  S r  r^  rA  s       r/   r   zrdist_gen._rvs  s,    <$$QqS!A#t444q88r1   c                     d|dz  z
  t          j        |dz   dz  |dz            z  }|t          j        d|dz            z  S )Nr   rH   r|   r   r   r  )r:   rU   r  	numerators       r/   r   zrdist_gen._munp  sG    !a%[BGQWM1s7$C$CC	2761r62222r1   r   )rv   rw   rx   ry   r^   re   r   rh   rq   r   r   rz   r1   r/   r  r    s           BE E E* * *> > >. . ., , ,9 9 9 93 3 3 3 3r1   r  r  rdistc                        e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zd Zd Ze eed           fd                        Z xZS )rayleigh_gena7  A Rayleigh continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rayleigh` is:

    .. math::

        f(x) = x \exp(-x^2/2)

    for :math:`x \ge 0`.

    `rayleigh` is a special case of `chi` with ``df=2``.

    %(after_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zrayleigh_gen._shape_info  r   r1   Nc                 <    t                               d||          S )NrH   r  r&  r   s      r/   r   zrayleigh_gen._rvs  s    wwqt,w???r1   c                 P    t          j        |                     |                    S rA   r  r:   r  s     r/   re   zrayleigh_gen._pdf  r  r1   c                 <    t          j        |          d|z  |z  z
  S r_  r#  r  s     r/   r   zrayleigh_gen._logpdf  s    vayy37Q;&&r1   c                 8    t          j        d|dz  z             S r  r  r  s     r/   rh   zrayleigh_gen._cdf  s    1%%%%r1   c                 V    t          j        dt          j        |           z            S Nr  )rC   r   rj   r  r   s     r/   rq   zrayleigh_gen._ppf"  s!    wrBHaRLL()))r1   c                 P    t          j        |                     |                    S rA   r  r  s     r/   rl   zrayleigh_gen._sf%  s    vdkk!nn%%%r1   c                     d|z  |z  S )Nrq  rz   r  s     r/   r   zrayleigh_gen._logsf(  s    ax!|r1   c                 T    t          j        dt          j        |          z            S r  )rC   r   r   r   s     r/   rt   zrayleigh_gen._isf+  s    wrBF1II~&&&r1   c                    dt           j        z
  }t          j        t           j        dz            |dz  dt           j        dz
  z  t          j        t           j                  z  |dz  z  dt           j        z  |z  d|dz  z  z
  fS )Nr  rH   r  ri  r  r  r.  r/  s     r/   r   zrayleigh_gen._stats.  sp    "%ia  A257BGBENN*383"%BsAvI%' 	'r1   c                 L    t           dz  dz   dt          j        d          z  z
  S )Nr   r   r   rH   r  r]   s    r/   r   zrayleigh_gen._entropy5  s!    czA~BF1II--r1   a          Notes specifically for ``rayleigh.fit``: If the location is fixed with
        the `floc` parameter, this method uses an analytical formula to find
        the scale.  Otherwise, this function uses a numerical root finder on
        the first order conditions of the log-likelihood function to find the
        MLE.  Only the (optional) `loc` parameter is used as the initial guess
        for the root finder; the `scale` parameter and any other parameters
        for the optimizer are ignored.

r   c                    |                     dd          r t                      j        g|R i |S t          | ||          \  }}fd}fd}|ffd	}|Dt	          j        |z
  dk              rt          ddt          j        	          | ||          fS |                    d
          }	|	| 	                              d         }	||n|}
t	          j
        t	          j                  t          j                   }t          |
|          }t          j        |
||f          }|j        st!          |j                  |j        }|p
 ||          }||fS )Nr  Fc                 d    t          j        | z
  dz            dt                    z  z  dz  S ra  )rC   r  r  )r'   r   s    r/   	scale_mlez#rayleigh_gen.fit.<locals>.scale_mleG  s2     FD3J1,--SYY?BFFr1   c                     | z
  }|                                 }|dz                                   }d|z                                   }||dt                    z  z  |z  z
  S r  )r  r  )r'   r&  rM  rS  s3r   s        r/   loc_mlez!rayleigh_gen.fit.<locals>.loc_mleL  s\     BBa%BB$BAc$iiK(+++r1   c                 r    | z
  }|                                 |dz  d|z                                   z  z
  S r  )r  )r'   r(   r&  r   s      r/   loc_mle_scale_fixedz-rayleigh_gen.fit.<locals>.loc_mle_scale_fixedU  s6     B6688eQh!B$555r1   r   rayleighr   r  r'   r`  )r,   r7   r9   r  rC   r  r<  r\   r5   r  rb  r  rM   r   r$   ra  rJ   flagr  )r:   r   r;   r.   r   r   r  r  r  loc0r<   rF   rE   r  r'   r(   r  s    `              r/   r9   zrayleigh_gen.fit8  s    88J&& 	4577;t3d333d3338t9=tE EdF	G 	G 	G 	G 	G
	, 	, 	, 	, 	, ,2 	6 	6 	6 	6 	6 	6 vdTkQ&'' -":QbfEEEEYYt__,, xx<>>$''*Dgg-@bfTllRVG44"3//"30@AAA} 	+ ***h())C..Ezr1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   r   rh   rq   rl   r   rt   r   r   r>   r   r9   r  r  s   @r/   r  r    s,        * "4M  @ @ @ @' ' '' ' '& & &* * *& & &  ' ' '' ' '. . . } 5. / / // / / // / _/ / / / /r1   r  r  c                        e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd ZdZ eee           fd            Z xZS )reciprocal_gena  A loguniform or reciprocal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for this class is:

    .. math::

        f(x, a, b) = \frac{1}{x \log(b/a)}

    for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
    :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    This doesn't show the equal probability of ``0.01``, ``0.1`` and
    ``1``. This is best when the x-axis is log-scaled:

    >>> import numpy as np
    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log10(r))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    This random variable will be log-uniform regardless of the base chosen for
    ``a`` and ``b``. Let's specify with base ``2`` instead:

    >>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000)

    Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random
    variable.  Here's the histogram:

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log2(rvs))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    c                     |dk    ||k    z  S r:  rz   r|  s      r/   rV   zreciprocal_gen._argcheck      A!a%  r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS rX  r[   rY  s      r/   r^   zreciprocal_gen._shape_info  r\  r1   c                     t                                          |t          j        |          t          j        |          f          S Nr  r7   r  rC   r  r  r  s     r/   r  zreciprocal_gen._fitstart  5    ww  RVD\\26$<<,H IIIr1   c                 
    ||fS rA   rz   r|  s      r/   r   zreciprocal_gen._get_support  r  r1   c                 B    d|t          j        |dz  |z            z  z  S r  r#  rc  s       r/   re   zreciprocal_gen._pdf  s$    a"&S1---..r1   c                     t          j        |           t          j        t          j        |dz  |z                      z
  S r  r#  rc  s       r/   r   zreciprocal_gen._logpdf  s3    q		zBF26!c'A+#6#67777r1   c                     t          j        |          t          j        |          z
  t          j        |dz  |z            z  S r  r#  rc  s       r/   rh   zreciprocal_gen._cdf  s4    q		"&))#rva#gk':':::r1   c                 4    |t          |dz  |z  |          z  S r  r  rN  s       r/   rq   zreciprocal_gen._ppf  s    QsU1Wa  r1   c                     dt          j        |dz  |z            z  |z  t          |dz  |          t          |dz  |          z
  z  S r  r  r  s       r/   r   zreciprocal_gen._munp  sD    26!C%'??"Q&#aeQ--#aeQ--*GHHr1   c                     dt          j        ||z            z  t          j        t          j        |dz  |z                      z   S r   r#  r|  s      r/   r   zreciprocal_gen._entropy  s7    26!A#;;rvbfQsU1Woo6666r1   z        `loguniform`/`reciprocal` is over-parameterized. `fit` automatically
         fixes `scale` to 1 unless `fscale` is provided by the user.

r   c                 n    |                     dd          } t                      j        |g|R d|i|S )Nr   r   )r,   r7   r9   )r:   r   r;   r.   r   r  s        r/   r9   zreciprocal_gen.fit  sA    (A&&uww{4>$>>>v>>>>r1   )rv   rw   rx   ry   rV   r^   r  r   re   r   rh   rq   r   r   fit_noter   r   r9   r  r  s   @r/   r  r  v  s       1 1d! ! !  
J J J J J  / / /8 8 8; ; ;! ! !I I I7 7 7LH }H===? ? ? ? >=? ? ? ? ?r1   r  
loguniform
reciprocalc                   >    e Zd ZdZd Zd Zd
dZd Zd Zd Z	d	 Z
dS )rice_gena  A Rice continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rice` is:

    .. math::

        f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)

    for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
    function of order zero (`scipy.special.i0`).

    `rice` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    The Rice distribution describes the length, :math:`r`, of a 2-D vector with
    components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
    v` are independent Gaussian random variables with standard deviation
    :math:`s`.  Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
    ``rice.pdf(x, R/s, scale=s)``.

    %(example)s

    c                     |dk    S r:  rz   r:   r   s     r/   rV   zrice_gen._argcheck  r;  r1   c                 @    t          dddt          j        fd          gS )Nr   Fr   rZ   r[   r]   s    r/   r^   zrice_gen._shape_info  r>  r1   Nc                     |t          j        d          z  |                    d|z             z   }t          j        ||z                      d                    S )NrH   )rH   r  r   rM  )rC   r   r   r  )r:   r   r   r   r$  s        r/   r   zrice_gen._rvs  sO    bgajjL<77TD[7IIIw!yyay(()))r1   c                 v    t          j        t          j        |          dt          j        |                    S r	  )rj   chndtrrC   squarer  s      r/   rh   zrice_gen._cdf  s&    y1q")A,,777r1   c           	      v    t          j        t          j        |dt          j        |                              S r	  )rC   r   rj   chndtrixr	  r  s      r/   rq   zrice_gen._ppf  s(    wr{1a166777r1   c                 z    |t          j        ||z
   ||z
  z  dz            z  t          j        ||z            z  S r0  )rC   r   rj   i0er  s      r/   re   zrice_gen._pdf  s=     26AaC&!A#,s*+++bfQqSkk99r1   c                     |dz  }d|z   }||z  dz  }d|z  t          j        |           z  t          j        |          z  t          j        |d|          z  S r  )rC   r   rj   r  r  )r:   rU   r   nd2n1r  s         r/   r   zrice_gen._munp  s_    eWqSWc
RVRC[[(28B<<7	"a$$% 	&r1   r   )rv   rw   rx   ry   rV   r^   r   rh   rq   re   r   rz   r1   r/   r 	  r 	    s         8  D D D* * * *
8 8 88 8 8: : :& & & & &r1   r 	  ricec                   8    e Zd ZdZd Zd Zd Zd Zd Zd	dZ	dS )
recipinvgauss_gena  A reciprocal inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `recipinvgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
                    \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)

    for :math:`x \ge 0`.

    `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   zrecipinvgauss_gen._shape_info2  rP  r1   c                 R    t          j        |                     ||                    S rA   r  r  s      r/   re   zrecipinvgauss_gen._pdf5  s"     vdll1b))***r1   c                 L    t          |dk    ||fd t          j                   S )Nr   c                     d|| z  z
  dz   d| z  |dz  z  z  dt          j        dt           j        z  | z            z  z
  S )Nr   r   rH   r   r   )rd   r5  s     r/   r  z+recipinvgauss_gen._logpdf.<locals>.<lambda><  sH    1r!t8c/)9QqSS[)I+.rvagai/@/@+@*A r1   r  r  r  s      r/   r   zrecipinvgauss_gen._logpdf:  s8    !a%!RB B%'VG- - - 	-r1   c                     d|z  |z
  }d|z  |z   }dt          j        |          z  }t          | |z            t          j        d|z            t          | |z            z  z
  S Nr|   r   rC   r   r   r   r:   rd   r5  r-  r/  isqxs         r/   rh   zrecipinvgauss_gen._cdf@  se    2vz2vz271::~$t$$rvc"f~~id
6K6K'KKKr1   c                     d|z  |z
  }d|z  |z   }dt          j        |          z  }t          ||z            t          j        d|z            t          | |z            z  z   S r	  r	  r	  s         r/   rl   zrecipinvgauss_gen._sfF  sc    2vz2vz271::~d##bfSVnnYuTz5J5J&JJJr1   Nc                 8    d|                     |d|          z  S r  r  r  s       r/   r   zrecipinvgauss_gen._rvsL  s"    <$$R4$8888r1   r   )
rv   rw   rx   ry   r^   re   r   rh   rl   r   rz   r1   r/   r	  r	    s         ,F F F+ + +
- - -L L LK K K9 9 9 9 9 9r1   r	  recipinvgaussc                   D    e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
d
 ZdS )semicircular_gena  A semicircular continuous random variable.

    %(before_notes)s

    See Also
    --------
    rdist

    Notes
    -----
    The probability density function for `semicircular` is:

    .. math::

        f(x) = \frac{2}{\pi} \sqrt{1-x^2}

    for :math:`-1 \le x \le 1`.

    The distribution is a special case of `rdist` with `c = 3`.

    %(after_notes)s

    References
    ----------
    .. [1] "Wigner semicircle distribution",
           https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zsemicircular_gen._shape_infor  r   r1   c                 V    dt           j        z  t          j        d||z  z
            z  S r  r.  r   s     r/   re   zsemicircular_gen._pdfu  s#    25y1Q3''r1   c                 |    t          j        dt           j        z            dt          j        | |z            z  z   S ra  r8  r   s     r/   r   zsemicircular_gen._logpdfx  s.    vagRXqbd^^!333r1   c                     ddt           j        z  |t          j        d||z  z
            z  t          j        |          z   z  z   S )Nr   r|   r   )rC   r   r   r  r   s     r/   rh   zsemicircular_gen._cdf{  s:    3ru9a!A#.1=>>>r1   c                 8    t                               |d          S Nr  )r  rq   r   s     r/   rq   zsemicircular_gen._ppf~  s    zz!Qr1   Nc                     t          j        |                    |                    }t          j        t           j        |                    |          z            }||z  S rO  )rC   r   r  r  r   )r:   r   r   r  r~   s        r/   r   zsemicircular_gen._rvs  sT     GL((d(3344F25<//T/:::;;1ur1   c                     dS )N)r   r*  r   r  rz   r]   s    r/   r   zsemicircular_gen._stats  r  r1   c                     dS )NgzCϑ?rz   r]   s    r/   r   zsemicircular_gen._entropy  s    %%r1   r   )rv   rw   rx   ry   r^   re   r   rh   rq   r   r   r   rz   r1   r/   r	  r	  S  s         <  ( ( (4 4 4? ? ?             & & & & &r1   r	  semicircularc                   >    e Zd ZdZd Zd Zd Zd Zd ZddZ	d	 Z
d
S )skewcauchy_gena  A skewed Cauchy random variable.

    %(before_notes)s

    See Also
    --------
    cauchy : Cauchy distribution

    Notes
    -----

    The probability density function for `skewcauchy` is:

    .. math::

        f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1
                                                   \right)^2} + 1 \right)}

    for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`.

    When :math:`a=0`, the distribution reduces to the usual Cauchy
    distribution.

    %(after_notes)s

    References
    ----------
    .. [1] "Skewed generalized *t* distribution", Wikipedia
       https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution

    %(example)s

    c                 2    t          j        |          dk     S rQ   )rC   r  r  s     r/   rV   zskewcauchy_gen._argcheck  s    vayy1}r1   c                 (    t          dddd          gS )Nr~   F)r  r|   r   r   r]   s    r/   r^   zskewcauchy_gen._shape_info  s    3{NCCDDr1   c                 n    dt           j        |dz  |t          j        |          z  dz   dz  z  dz   z  z  S r"  )rC   r   rD   r  s      r/   re   zskewcauchy_gen._pdf  s8    BEQTQ^a%7!$;;a?@AAr1   c                    t          j        |dk    d|z
  dz  d|z
  t           j        z  t          j        |d|z
  z            z  z   d|z
  dz  d|z   t           j        z  t          j        |d|z   z            z  z             S Nr   r   rH   )rC   r  r   r4  r  s      r/   rh   zskewcauchy_gen._cdf  s    xQQ!q1uo	!q1u+8N8N&NNQ!q1uo	!q1u+8N8N&NNP P 	Pr1   c           
      2   ||                      d|          k     }t          j        |t          j        t          j        d|z
  z  |d|z
  dz  z
  z            d|z
  z  t          j        t          j        d|z   z  |d|z
  dz  z
  z            d|z   z            S r1	  )rh   rC   r  r8  r   )r:   rd   r~   r  s       r/   rq   zskewcauchy_gen._ppf  s    		!QxruA!q1uk/BCCq1uMruA!q1uk/BCCq1uMO O 	Or1   rz  c                 ^    t           j        t           j        t           j        t           j        fS rA   r<  )r:   r~   r  s      r/   r   zskewcauchy_gen._stats  r=  r1   c                 N    t          j        |g d          \  }}}d|||z
  dz  fS )NrA  r{   rH   rE  )r:   r   rG  rH  rI  s        r/   r  zskewcauchy_gen._fitstart  s4     dLLL99S#C#)Q&&r1   Nr  )rv   rw   rx   ry   rV   r^   re   rh   rq   r   r  rz   r1   r/   r+	  r+	    s           B  E E EB B BP P P
O O O. . . .' ' ' ' 'r1   r+	  
skewcauchyc                        e Zd ZdZd Zd Zd Z fdZd Zd Z	d Z
dd
ZddZed             Zd Z eed           fd            Z xZS )skew_norm_genaf  A skew-normal random variable.

    %(before_notes)s

    Notes
    -----
    The pdf is::

        skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)

    `skewnorm` takes a real number :math:`a` as a skewness parameter
    When ``a = 0`` the distribution is identical to a normal distribution
    (`norm`). `rvs` implements the method of [1]_.

    %(after_notes)s

    %(example)s

    References
    ----------
    .. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of
        the multivariate skew-normal distribution. J. Roy. Statist. Soc.,
        B 61, 579-602. :arxiv:`0911.2093`

    c                 *    t          j        |          S rA   r  r  s     r/   rV   zskew_norm_gen._argcheck  r  r1   c                 V    t          ddt          j         t          j        fd          gS )Nr~   Fr   r[   r]   s    r/   r^   zskew_norm_gen._shape_info  r  r1   c                 8    t          |dk    ||fd d           S )Nr   c                      t          |           S rA   r   rd   r~   s     r/   r  z$skew_norm_gen._pdf.<locals>.<lambda>  s    1 r1   c                 L    dt          |           z  t          || z            z  S r0  r  r<	  s     r/   r  z$skew_norm_gen._pdf.<locals>.<lambda>  s    By||OIacNN: r1   r  r  r  s      r/   re   zskew_norm_gen._pdf  s2    FQF55::
 
 
 	
r1   c                    t          j        |dd|          }t          j        ||j                  }|dk     |dk    z  }t                                          ||         ||                   ||<   t          j        |dd          S )Nr   r   gư>)ra  _skewnorm_cdfrC   r  r  r7   rh   r,  )r:   rd   r~   r   i_small_cdfr  s        r/   rh   zskew_norm_gen._cdf  sv    "1aA..OAsy))Tza!e, 77<<++GGKwsAq!!!r1   c                 0    t          j        |dd|          S r1  )ra  _skewnorm_ppfr  s      r/   rq   zskew_norm_gen._ppf       #Aq!Q///r1   c                 2    |                      | |           S rA   rb  r  s      r/   rl   zskew_norm_gen._sf   s     yy!aR   r1   c                 0    t          j        |dd|          S r1  )ra  _skewnorm_isfr  s      r/   rt   zskew_norm_gen._isf   rC	  r1   Nc                    |                     |          }|                     |          }|t          j        d|dz  z             z  }||z  |t          j        d|dz  z
            z  z   }t          j        |dk    ||           S )Nr  r   rH   r   )normalrC   r   r  )r:   r~   r   r   u0r  r  r  s           r/   r   zskew_norm_gen._rvs   s      d ++T**bga!Q$hrTAbga!Q$h''''xabS)))r1   rz  c                    g d}t          j        dt           j        z            |z  t          j        d|dz  z             z  }d|v r||d<   d|v rd|dz  z
  |d<   d|v r6dt           j        z
  dz  |t          j        d|dz  z
            z  d	z  z  |d<   d
|v r'dt           j        d	z
  z  |dz  d|dz  z
  dz  z  z  |d	<   |S )NrI  rH   r   rT  r   r  r  r  r  r  rX  )r:   r~   r  r  consts        r/   r   zskew_norm_gen._stats   s    )))"%  1$RWQAX%6%66'>>F1I'>>E1HF1I'>>be)Q5UAX1F1F+F*JJF1I'>>BEAI5!8Q\A4E+EFF1Ir1   c                 J   t          dg          t          ddg          t          g d          t          g d          t          g d          t          g d          t          g d          t          g d	          t          g d
          t          g d          d
}|S )Nr   r  r  )r  ir  )i   i?   i)i  iin  irM	  )(  iSi6Q  ii  iO)i iBi/ iio irO	  ) iԷi iYei{Hx ii i!)	i!iׅi쇀iiViX'ilirP	  )
is_'il   </1 ldy( l   J8D l.~ l   -Rx iWi[i0)
r   r  r  rH  r  r1     r        r   )r:   skewnorm_odd_momentss     r/   _skewnorm_odd_momentsz#skew_norm_gen._skewnorm_odd_moments,   s     1#1b'"",,,''...//77788EEEFF # # # $ $ 8 8 8 9 9 % % % & &  2 2 2 3 3 
  
$ $#r1   c                    |dz  rV|dk    rt          d          |t          j        d|dz  z             z  }| | j        |         |dz            z  t          z  S t          j        |dz   dz            d|dz  z  z  t          z  S )Nr   rS	  zKskewnorm noncentral moments not implemented for odd orders greater than 19.rH   )r  rC   r   rU	  r"   rj   r  r!   )r:   orderr~   r(  s       r/   r   zskew_norm_gen._munpB   s    19 	Erzz) +5 6 6 6
 bga!Q$h'''E=D6u=eQhGGG%& ' 8UQYM**Qq\9HDDr1   a          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        Note that the maximum possible skewness magnitude of a
        `scipy.stats.skewnorm` distribution is approximately 0.9952717; if the
        magnitude of the data's sample skewness exceeds this, the returned
        shape parameter ``a`` will be infinite.
        

r   c           	         t          | |||          \  }}}}|                    dd                                          }d t          j        |           d          }t                    |k    r'|dk    r!||s t                      j        |g|R i |S |dk    rd\  }	}
}nEt          |          r|d         nd }	|	                    dd           }
|	                    d	d           }||	t          j        | |          t          fd
ddg          j        }t          j        d          5  t          j        t          j        |dz  d|dz  z
                      t          j                  z  }	d d d            n# 1 swxY w Y   n#||n|	}	|	t          j        d|	dz  z             z  }|D|Bt          j        |          }t          j        |dd|dz  z  t          j        z  z
  z            }n||}|A|
?t          j        |          }|||z  t          j        dt          j        z            z  z
  }
n||}
|dk    r|	|
|fS  t                      j        ||	f|
|d|S )Nr*   r4   c                     dt           j        z
  dz  | t          j        dt           j        z            z  dz  dd| dz  z  t           j        z  z
  dz  z  z  S )Nr  rH   r  r   ri  r.  r  s    r/   skew_dz!skew_norm_gen.fit.<locals>.skew_dk   sX    beGQ;1rwq25y'9'9#9A"=%&1a4"%%73$?#@ A Ar1   r   r  r  r   r'   r(   c                       |           z
  S rA   rz   )r  r  r[	  s    r/   r  z#skew_norm_gen.fit.<locals>.<lambda>   s    ffQii!m r1   r  r`  r+  r,  rH   r  )r  r5   r6   r  r  r  r7   r9   r  r,   rC   r,  r$   r  r.  r   r-  rD   r  r   r   )r:   r   r;   r.   r  r   r   r*   s_maxr~   r'   r(   r  r  rT  r  r[	  r  s                  @@r/   r9   zskew_norm_gen.fitV   s    "=T4=A4"I "Ib$(E**0022	A 	A 	A Jtq		q66U??v~~"*T*577;t3d333d333 T>>,MAsEEt99.Q$A((5$''CHHWd++E:!) E65))A33333b!WEEEJAH--- B BGBIadQq!tV5566rwqzzAB B B B B B B B B B B B B B B n!ABGA1H%%%A>emtAGAQq!tVBE\!1233EEE<CKAeAgbgag....CCCT>>c5=  577;tQECuEEEEEs   =AFFFr   r  )rv   rw   rx   ry   rV   r^   re   rh   rq   rl   rt   r   r   r   rU	  r   r   r   r9   r  r  s   @r/   r7	  r7	    s>        2  K K K
 
 
" " " " "0 0 0! ! !
0 0 0* * * *   * $ $ _$*E E E( } 5   9F 9F 9F 9F 9F 9F 9F 9F 9Fr1   r7	  skewnormc                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
trapezoid_gena  A trapezoidal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The trapezoidal distribution can be represented with an up-sloping line
    from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
    and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``.  This
    defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat
    top from ``c`` to ``d`` proportional to the position along the base
    with ``0 <= c <= d <= 1``.  When ``c=d``, this is equivalent to `triang`
    with the same values for `loc`, `scale` and `c`.
    The method of [1]_ is used for computing moments.

    `trapezoid` takes :math:`c` and :math:`d` as shape parameters.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    References
    ----------
    .. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular
       distributions for Type B evaluation of standard uncertainty.
       Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003`


    c                 F    |dk    |dk    z  |dk    z  |dk    z  ||k    z  S r1  rz   r:   r  r  s      r/   rV   ztrapezoid_gen._argcheck   s0    Q16"a1f-a8AFCCr1   c                 R    t          dddd          }t          dddd          }||gS )Nr  Fr   r|   TTr  r.	  r  s      r/   r^   ztrapezoid_gen._shape_info   s1    UHl;;UHl;;Bxr1   c                 z    d||z
  dz   z  }t          ||k     ||k    ||k    z  ||k    gd d d g||||f          S )NrH   r   c                     || z  |z  S rA   rz   rd   r  r  r  s       r/   r  z$trapezoid_gen._pdf.<locals>.<lambda>   s    q1uqy r1   c                     |S rA   rz   rh	  s       r/   r  z$trapezoid_gen._pdf.<locals>.<lambda>   s    q r1   c                     |d| z
  z  d|z
  z  S rQ   rz   rh	  s       r/   r  z$trapezoid_gen._pdf.<locals>.<lambda>   s    qAaCyAaC/@ r1   r   )r:   rd   r  r  r  s        r/   re   ztrapezoid_gen._pdf   sn    1QKAE!VQ/E# 9800@@B 1aL* * 	*r1   c                 b    t          ||k     ||k    ||k    z  ||k    gd d d g|||f          S )Nc                 $    | dz  |z  ||z
  dz   z  S r  rz   rd   r  r  s      r/   r  z$trapezoid_gen._cdf.<locals>.<lambda>   s    AqD1H!A,> r1   c                 *    |d| |z
  z  z   ||z
  dz   z  S r  rz   rn	  s      r/   r  z$trapezoid_gen._cdf.<locals>.<lambda>   s    Qac]qs1u,E r1   c                 6    dd| z
  dz  ||z
  dz   z  d|z
  z  z
  S r"  rz   rn	  s      r/   r  z$trapezoid_gen._cdf.<locals>.<lambda>   s3    A!z23A#a%09<=aC0A -B r1   rk	  r  s       r/   rh   ztrapezoid_gen._cdf   sa    AE!VQ/E# ?>EEB BC 1I' ' 	'r1   c                 b   |                      |||          |                      |||          }}||k     ||k    ||k    g}t          j        ||z  d|z   |z
  z            d|z  d|z   |z
  z  d|z  z   dt          j        d|z
  ||z
  dz   z  d|z
  z            z
  g}t          j        ||          S r  )rh   rC   r   select)r:   rp   r  r  qcqdrZ  r"  s           r/   rq   ztrapezoid_gen._ppf   s    1a##TYYq!Q%7%7BFAGQV,ga!eq1uqy122AgQ+cAg5"'1q5QUQY"71q5"ABBBD
 y:...r1   c                     |dz   z  }t          |dk    d|k     |dk     z  |dk    gd fdfdg|g          }dd|z   |z
  z  ||z
  z  dz   dz   z  z  }|S )	Nr   r{   r|   c                     dS r  rz   rZ	  s    r/   r  z%trapezoid_gen._munp.<locals>.<lambda>   s    s r1   c                 h    t          j        dz   t          j        |           z            | dz
  z  S rU  )rC   r  r   r  rU   s    r/   r  z%trapezoid_gen._munp.<locals>.<lambda>   s+    rx1q		 122ae< r1   c                     dz   S r	  rz   rx	  s    r/   r  z%trapezoid_gen._munp.<locals>.<lambda>   s    qs r1   r   rH   rk	  )r:   rU   r  r  ab_termdc_termr  s    `     r/   r   ztrapezoid_gen._munp   s     ac(#XaAG,a3h7]<<<<]]] C  SU1Wo7!23!!}E
r1   c                 f    dd|z
  |z   z  d|z   |z
  z  t          j        dd|z   |z
  z            z   S r   r#  rb	  s      r/   r   ztrapezoid_gen._entropy!  s>     c!eAg#a%'*RVC3q57O-D-DDDr1   N)rv   rw   rx   ry   rV   r^   re   rh   rq   r   r   rz   r1   r/   r`	  r`	     s           BD D D  
	* 	* 	*' ' '/ / /  2E E E E Er1   r`	  	trapezoidtrapzz!trapz is an alias for `trapezoid`c                   D    e Zd ZdZddZd Zd Zd Zd Zd Z	d	 Z
d
 ZdS )
triang_gena5  A triangular continuous random variable.

    %(before_notes)s

    Notes
    -----
    The triangular distribution can be represented with an up-sloping line from
    ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
    to ``(loc + scale)``.

    `triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    Nc                 2    |                     d|d|          S r1  )
triangularrA  s       r/   r   ztriang_gen._rvs)!  s    &&q!Q555r1   c                     |dk    |dk    z  S r1  rz   r)  s     r/   rV   ztriang_gen._argcheck,!  s    Q16""r1   c                 (    t          dddd          gS )Nr  Frd	  re	  r.	  r]   s    r/   r^   ztriang_gen._shape_info/!  s    3x>>??r1   c                 r    t          |dk    ||k     ||k    |dk    z  |dk    gd d d d g||f          }|S )Nr   r   c                     dd| z  z
  S r	  rz   r  s     r/   r  z!triang_gen._pdf.<locals>.<lambda><!  s    a!a%i r1   c                     d| z  |z  S r	  rz   r  s     r/   r  z!triang_gen._pdf.<locals>.<lambda>=!      a!eai r1   c                     dd| z
  z  d|z
  z  S r  rz   r  s     r/   r  z!triang_gen._pdf.<locals>.<lambda>>!  s    a1q5kQU&; r1   c                     d| z  S r	  rz   r  s     r/   r  z!triang_gen._pdf.<locals>.<lambda>?!  
    a!e r1   rk	  r:   rd   r  r  s       r/   re   ztriang_gen._pdf2!  sk     aQq&Q!V,a! 0///;;++- A    r1   c                 r    t          |dk    ||k     ||k    |dk    z  |dk    gd d d d g||f          }|S )Nr   r   c                     d| z  | | z  z
  S r	  rz   r  s     r/   r  z!triang_gen._cdf.<locals>.<lambda>H!  s    acAaCi r1   c                     | | z  |z  S rA   rz   r  s     r/   r  z!triang_gen._cdf.<locals>.<lambda>I!  r	  r1   c                 *    | | z  d| z  z
  |z   |dz
  z  S r  rz   r  s     r/   r  z!triang_gen._cdf.<locals>.<lambda>J!  s    qsQqSy1}1&= r1   c                     | | z  S rA   rz   r  s     r/   r  z!triang_gen._cdf.<locals>.<lambda>K!  r	  r1   rk	  r	  s       r/   rh   ztriang_gen._cdfC!  si    aQq&Q!V,a! 0///==++- A    r1   c           
          t          j        ||k     t          j        ||z            dt          j        d|z
  d|z
  z            z
            S rQ   )rC   r  r   r  s      r/   rq   ztriang_gen._ppfO!  sA    xArwq1u~~q!A#!A#1G1G/GHHHr1   c           	          |dz   dz  d|z
  ||z  z   dz  t          j        d          d|z  dz
  z  |dz   z  |dz
  z  dt          j        d|z
  ||z  z   d          z  z  dfS )	Nr|   r     rH   r   r  ri  g333333)rC   r   rj  r)  s     r/   r   ztriang_gen._statsR!  sy    3QqsB

AaCE"AaC(!A#.!BHc!eAaCi#4N4N2NO 	r1   c                 0    dt          j        d          z
  S rZ  r#  r)  s     r/   r   ztriang_gen._entropyX!  s    26!99}r1   r   )rv   rw   rx   ry   r   rV   r^   re   rh   rq   r   r   rz   r1   r/   r	  r	  !  s         *6 6 6 6# # #@ @ @  "
 
 
I I I      r1   r	  triangc                   B    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
S )truncexpon_genad  A truncated exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `truncexpon` is:

    .. math::

        f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}

    for :math:`0 <= x <= b`.

    `truncexpon` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS r  r[   r]   s    r/   r^   ztruncexpon_gen._shape_infou!  r   r1   c                     | j         |fS rA   r  r	  s     r/   r   ztruncexpon_gen._get_supportx!      vqyr1   c                 Z    t          j        |           t          j        |            z  S rA   r  r  s      r/   re   ztruncexpon_gen._pdf{!  s#    vqbzzBHaRLL=))r1   c                 Z    | t          j        t          j        |                      z
  S rA   r  r  s      r/   r   ztruncexpon_gen._logpdf!  s%    rBFBHaRLL=))))r1   c                 X    t          j        |           t          j        |           z  S rA   r  r  s      r/   rh   ztruncexpon_gen._cdf!  s!    x||BHaRLL((r1   c                 X    t          j        |t          j        |           z             S rA   )rj   r  r  r  s      r/   rq   ztruncexpon_gen._ppf!  s#    28QB<<((((r1   c                 8   |dk    r5d|dz   t          j        |           z  z
  t          j        |            z  S |dk    rDddd||z  d|z  z   dz   z  t          j        |           z  z
  z  t          j        |            z  S |                     ||          S r  )rC   r   rj   r  r~  )r:   rU   r   s      r/   r   ztruncexpon_gen._munp!  s     66qsBFA2JJ&&"(A2,,77!VVaQqS1WQYr

223bhrll]CC ==A&&&r1   c                 |    t          j        |          }t          j        |dz
            d||dz
  z  z   d|z
  z  z   S r  )rC   r   r   )r:   r   eBs      r/   r   ztruncexpon_gen._entropy!  s;    VAYYvbd||Qr1S5z\CF333r1   N)rv   rw   rx   ry   r^   r   re   r   rh   rq   r   r   rz   r1   r/   r	  r	  _!  s         *E E E  * * ** * *) ) )) ) )
' 
' 
'4 4 4 4 4r1   r	  
truncexponc                 2    t          j        | |gd          S )Nr   rM  )rj   r  log_plog_qs     r/   _log_sumr	  !  s    <Q////r1   c                 R    t          j        | |t          j        dz  z   gd          S )Ny              ?r   rM  )rj   r  rC   r   r	  s     r/   	_log_diffr	  !  s&    <beBh/a8888r1   c                 ,   t          j        |           t          j        |          }} t          j        | |          \  } }|dk    }| dk    }||z   }d fd}d }t          j        | t           j        t           j                  }| |         j        r | |         ||                   ||<   | |         j        r || |         ||                   ||<   | |         j        r || |         ||                   ||<   t          j        |          S )z3Log of Gaussian probability mass within an intervalr   c                 j    t          t          j        |          t          j        |                     S rA   )r	  rj   r   r~   r   s     r/   mass_case_leftz'_log_gauss_mass.<locals>.mass_case_left!  s"    QQ888r1   c                       | |            S rA   rz   )r~   r   r	  s     r/   mass_case_rightz(_log_gauss_mass.<locals>.mass_case_right!  s    ~qb1"%%%r1   c                 |    t          j        t          j        |            t          j        |           z
            S rA   )rj   r  r   r	  s     r/   mass_case_centralz*_log_gauss_mass.<locals>.mass_case_central!  s-     xbgqbkk1222r1   )rF  r  )rC   r  r  r  r  
complex128r   real)	r~   r   	case_left
case_rightcase_centralr	  r	  r  r	  s	           @r/   _log_gauss_massr	  !  sA   =R]1--qAq!$$DAq QIQJ+,L9 9 9& & & & &3 3 3 ,qRV2=
A
A
AC| D')a	lCCI} H)/!J-:GGJ P--aoqOOL73<<r1   c                   r     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd ZddZ xZS )truncnorm_gena^  A truncated normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    This distribution is the normal distribution centered on ``loc`` (default
    0), with standard deviation ``scale`` (default 1), and clipped at ``a``,
    ``b`` standard deviations to the left, right (respectively) from ``loc``.
    If ``myclip_a`` and ``myclip_b`` are clip values in the sample space (as
    opposed to the number of standard deviations) then they can be converted
    to the required form according to::

        a, b = (myclip_a - loc) / scale, (myclip_b - loc) / scale

    %(example)s

    c                     ||k     S rA   rz   r|  s      r/   rV   ztruncnorm_gen._argcheck!  s    1ur1   c                     t          ddt          j         t          j        fd          }t          ddt          j         t          j        fd          }||gS )Nr~   FrZ   r   )FTr[   rY  s      r/   r^   ztruncnorm_gen._shape_info!  sG    UbfWbf$5}EEUbfWbf$5}EEBxr1   c                     t                                          |t          j        |          t          j        |          f          S r  r  r  s     r/   r  ztruncnorm_gen._fitstart!  r  r1   c                 
    ||fS rA   rz   r|  s      r/   r   ztruncnorm_gen._get_support!  r  r1   c                 T    t          j        |                     |||                    S rA   r  rc  s       r/   re   ztruncnorm_gen._pdf!  r  r1   c                 B    t          |          t          ||          z
  S rA   )r   r	  rc  s       r/   r   ztruncnorm_gen._logpdf!  s    AA!6!666r1   c                 T    t          j        |                     |||                    S rA   r0  rc  s       r/   rh   ztruncnorm_gen._cdf!  r  r1   c           
      R   t          j        |||          \  }}}t          ||          t          ||          z
  }|dk    }t          j        |          rQt          j        t          j        |                     ||         ||         ||                                        ||<   |S Ng)rC   r  r	  r  r  r   r   )r:   rd   r~   r   r  r  s         r/   r   ztruncnorm_gen._logcdf!  s    %aA..1a A&&A)>)>>TM6!99 	I"&QqT1Q41)F)F"G"G!GHHF1Ir1   c                 T    t          j        |                     |||                    S rA   r  rc  s       r/   rl   ztruncnorm_gen._sf"  r  r1   c           
      R   t          j        |||          \  }}}t          ||          t          ||          z
  }|dk    }t          j        |          rQt          j        t          j        |                     ||         ||         ||                                        ||<   |S r	  )rC   r  r	  r  r  r   r   )r:   rd   r~   r   r  r  s         r/   r   ztruncnorm_gen._logsf"  s    %aA..1a1%%1(=(==DL6!99 	IxQqT1Q41(F(F!G!G GHHE!Hr1   c                 ,   t          j        |||          \  }}}|dk     }| }d }d }t          j        |          }||         }	||         }
|	j        r ||	||         ||                   ||<   |
j        r ||
||         ||                   ||<   |S )Nr   c                     t          t          j        |          t          j        |           t          ||          z             }t          j        |          S rA   )r	  rj   r   rC   r   r	  	ndtri_exprp   r~   r   	log_Phi_xs       r/   ppf_leftz$truncnorm_gen._ppf.<locals>.ppf_left"  sG     Q!#_Q-B-B!BD DI<	***r1   c                     t          t          j        |           t          j        |            t          ||          z             }t          j        |           S rA   )r	  rj   r   rC   r  r	  r	  r	  s       r/   	ppf_rightz%truncnorm_gen._ppf.<locals>.ppf_right"  sN     aR!#1"10E0E!EG GIL++++r1   rC   r  
empty_liker   )r:   rp   r~   r   r	  r	  r	  r	  r  q_leftq_rights              r/   rq   ztruncnorm_gen._ppf"  s    %aA..1aE	Z
	+ 	+ 	+
	, 	, 	,
 mA9J-; 	J%Xfa	lAiLIIC	N< 	O'i:*NNC
O
r1   c                 ,   t          j        |||          \  }}}|dk     }| }d }d }t          j        |          }||         }	||         }
|	j        r ||	||         ||                   ||<   |
j        r ||
||         ||                   ||<   |S )Nr   c                     t          t          j        |          t          j        |           t          ||          z             }t          j        t          j        |                    S rA   )r	  rj   r   rC   r   r	  r	  r	  r	  s       r/   isf_leftz$truncnorm_gen._isf.<locals>.isf_left1"  sQ    !"+a.."$&))oa.C.C"CE EI<	 2 2333r1   c                     t          t          j        |           t          j        |            t          ||          z             }t          j        t          j        |                     S rA   )r	  rj   r   rC   r  r	  r	  r	  r	  s       r/   	isf_rightz%truncnorm_gen._isf.<locals>.isf_right6"  sX    !"+qb//"$(A2,,A1F1F"FH HIL!3!34444r1   r	  )r:   rp   r~   r   r	  r	  r	  r	  r  r	  r	  s              r/   rt   ztruncnorm_gen._isf*"  s    %aA..1aE	Z
	4 	4 	4
	5 	5 	5
 mA9J-; 	J%Xfa	lAiLIIC	N< 	O'i:*NNC
O
r1   c                       fd}t          |dk    ||k    z  ||k    z  |||ft          j        |t          j        g          t          j                  S )Nc                 0  	 
                     ||g||          \  }}|| g}ddg}t          d| dz             D ]T	t          ||||gg	fdd          }t          j        |          	dz
  |d         z  z   }|                    |           U|d         S )z
            Returns n-th moment. Defined only if n >= 0.
            Function cannot broadcast due to the loop over n
            r   r   c                     | |dz
  z  z  S rQ   rz   )rd   rV  r  s     r/   r  z:truncnorm_gen._munp.<locals>.n_th_moment.<locals>.<lambda>V"  s    q1qs8| r1   r  r  r  )re   r  r   rC   r  r  )rU   r~   r   pApBprobsr  r  mkr  r:   s            @r/   n_th_momentz(truncnorm_gen._munp.<locals>.n_th_momentH"  s    
 YY1vq!,,FB"IE!fG1ac]] # #
 "%%!Q";";";";qJ J JVD\\QqSGBK$77r""""2;r1   r   otypes)r   rC   r  float64r  )r:   rU   r~   r   r	  s   `    r/   r   ztruncnorm_gen._munpG"  sk    	 	 	 	 	& 16a1f-a81a),{BJ<HHH&" " 	"r1   r  c                     |                      t          j        ||g          ||          \  }}d }t          j        |d          } ||||||          S )Nc                 ^   ||z
  }|}|| g}t          ||| |ggd d          }dt          j        |          z   }	t          ||| |z
  ||z
  ggd d          }dt          j        |          z   }
t          ||| |ggd d          }d|z  t          j        |          z   }t          ||| |ggd d          }d	|	z  t          j        |          z   }||d
|	z  d|dz  z  z   z  z   }|t          j        |
d          z  }||d|z  d	|z  d|	z  |dz  z
  z  z   z  z   }||
dz  z  d	z
  }||
||fS )Nc                     | |z  S rA   rz   rU  s     r/   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>g"  s
    1Q3 r1   r   r  r   c                     | |z  S rA   rz   rU  s     r/   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>j"  s
    1 r1   c                     | |dz  z  S r	  rz   rU  s     r/   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>o"      1QT6 r1   rH   c                     | |dz  z  S r%	  rz   rU  s     r/   r  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>r"  r	  r1   r  rY  ri  rA  )r   rC   r  rj  )r~   r   r	  r	  r  r  r5  r	  r  r  r6  m3m4mu3r7  mu4r8  s                    r/   _truncnorm_stats_scalarz5truncnorm_gen._stats.<locals>._truncnorm_stats_scalarb"  s   bBB"IEeeaV_6F6F()+ + +DRVD\\!BeeadAbD\%:<L<L()+ + +D bfTll"CeeaV_6I6I()+ + +D2t$BeeaV_6I6I()+ + +D2t$BrRUQr1uW_--CrxS)))Br2b51R42A#6677CsAv!BsB?"r1   )r  )excluded)r   rC   r  r  )r:   r~   r   r  r	  r	  r	  _truncnorm_statss           r/   r   ztruncnorm_gen._stats_"  sp    "(Aq6**Aq11B	# 	# 	#4 <(?1=? ? ?1b"g666r1   r  )rv   rw   rx   ry   rV   r^   r  r   re   r   rh   r   rl   r   rq   rt   r   r   r  r  s   @r/   r	  r	  !  s        &    
J J J J J  - - -7 7 7- - -  , , ,    8  :" " "07 7 7 7 7 7 7 7r1   r	  	truncnorm)r   r   c                   f    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 Zd Zd Zd Zd Zd ZdS )truncpareto_genac  An upper truncated Pareto continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto : Pareto distribution

    Notes
    -----
    The probability density function for `truncpareto` is:

    .. math::

        f(x, b, c) = \frac{b}{1 - c^{-b}} \frac{1}{x^{b+1}}

    for :math:`b > 0`, :math:`c > 1` and :math:`1 \le x \le c`.

    `truncpareto` takes `b` and `c` as shape parameters for :math:`b` and
    :math:`c`.

    Notice that the upper truncation value :math:`c` is defined in
    standardized form so that random values of an unscaled, unshifted variable
    are within the range ``[1, c]``.
    If ``u_r`` is the upper bound to a scaled and/or shifted variable,
    then ``c = (u_r - loc) / scale``. In other words, the support of the
    distribution becomes ``(scale + loc) <= x <= (c*scale + loc)`` when
    `scale` and/or `loc` are provided.

    %(after_notes)s

    References
    ----------
    .. [1] Burroughs, S. M., and Tebbens S. F.
        "Upper-truncated power laws in natural systems."
        Pure and Applied Geophysics 158.4 (2001): 741-757.

    %(example)s

    c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )Nr   Fr{   r   r  r|   r[   )r:   r[  r  s      r/   r^   ztruncpareto_gen._shape_info"  s=    US"&M>BBUS"&M>BBBxr1   c                     |dk    |dk    z  S r1  rz   r:   r   r  s      r/   rV   ztruncpareto_gen._argcheck"  r  r1   c                     | j         |fS rA   r  r	  s      r/   r   ztruncpareto_gen._get_support"  r	  r1   c                 .    |||dz    z  z  d|| z  z
  z  S rQ   rz   r:   rd   r   r  s       r/   re   ztruncpareto_gen._pdf"  s%    1!f9}ArE	**r1   c                     t          j        |          t          j        || z             z
  |dz   t          j        |          z  z
  S rQ   r  r	  s       r/   r   ztruncpareto_gen._logpdf"  s<    vayy28QUF+++qsBF1IIo==r1   c                 (    d|| z  z
  d|| z  z
  z  S rQ   rz   r	  s       r/   rh   ztruncpareto_gen._cdf"  s!    ArE	a!aR%i((r1   c                 h    t          j        || z             t          j        || z             z
  S rA   r  r	  s       r/   r   ztruncpareto_gen._logcdf"  s1    xQB"(ArE6"2"222r1   c                 B    t          dd|| z  z
  |z  z
  d|z            S Nr   r  r  r:   rp   r   r  s       r/   rq   ztruncpareto_gen._ppf"  s)    1ArE	1}$bd+++r1   c                 0    || z  || z  z
  d|| z  z
  z  S rQ   rz   r	  s       r/   rl   ztruncpareto_gen._sf"  s'    A2A2!a!e),,r1   c                 t    t          j        || z  || z  z
            t          j        || z             z
  S rA   r  r	  s       r/   r   ztruncpareto_gen._logsf"  s9    va!ea!em$$rxQB'7'777r1   c                 J    t          || z  d|| z  z
  |z  z   d|z            S r	  r  r	  s       r/   rt   ztruncpareto_gen._isf"  s/    1qb5AA2Iq=("Q$///r1   c                     t          j        |d|| z  z
  z            |dz   t          j        |          ||z  dz
  z  d|z  z
  z  z    S rQ   r#  r	  s      r/   r   ztruncpareto_gen._entropy"  sV    1q1"u9&&aC"&))QTAX.1456 7 	7r1   c                     ||k    r!|t          j        |          z  d|| z  z
  z  S |||z
  z  ||z  ||z  z
  z  ||z  dz
  z  S rQ   r#  )r:   rU   r   r  s       r/   r   ztruncpareto_gen._munp"  sX    66RVAYY;!a!e),,!91q!t,1q99r1   c                 t    t                               |          \  }}}t          |          |z
  |z  }||||fS rA   )rQ  r9   r  )r:   r   r   r'   r(   r  s         r/   r  ztruncpareto_gen._fitstart"  s<    

4((3YY_e#!S%r1   N)rv   rw   rx   ry   r^   rV   r   re   r   rh   r   rq   rl   r   rt   r   r   r  rz   r1   r/   r	  r	  "  s        ' 'R  
! ! !  + + +> > >) ) )3 3 3, , ,- - -8 8 80 0 07 7 7: : :         r1   r	  truncparetoc                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
tukeylambda_gena*  A Tukey-Lamdba continuous random variable.

    %(before_notes)s

    Notes
    -----
    A flexible distribution, able to represent and interpolate between the
    following distributions:

    - Cauchy                (:math:`lambda = -1`)
    - logistic              (:math:`lambda = 0`)
    - approx Normal         (:math:`lambda = 0.14`)
    - uniform from -1 to 1  (:math:`lambda = 1`)

    `tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
    in the implementation) as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 *    t          j        |          S rA   r  r:   lams     r/   rV   ztukeylambda_gen._argcheck"  s    {3r1   c                 V    t          ddt          j         t          j        fd          gS )Nr

  Fr   r[   r]   s    r/   r^   ztukeylambda_gen._shape_info"  s$    5%26'26):NKKLLr1   c                 P   t          j        t          j        ||                    }||dz
  z  t          j        d|z
            |dz
  z  z   }dt          j        |          z  }t          j        |dk    t          |          dt          j        |          z  k     z  |d          S )Nr|   r   r   r{   )rC   r   rj   tklmbdar  r  )r:   rd   r

  Fxr  s        r/   re   ztukeylambda_gen._pdf#  s    Z
1c**++#c']bj2..#c'::Bxc!ffs2:c??/B&BCRMMMr1   c                 ,    t          j        ||          S rA   )rj   r
  )r:   rd   r

  s      r/   rh   ztukeylambda_gen._cdf#  s    z!S!!!r1   c                 Z    t          j        ||          t          j        | |          z
  S rA   )rj   r  r  )r:   rp   r

  s      r/   rq   ztukeylambda_gen._ppf
#  s'    yC  2;r3#7#777r1   c                 B    dt          |          dt          |          fS r:  )_tlvar_tlkurtr	
  s     r/   r   ztukeylambda_gen._stats#  s    &++q'#,,..r1   c                 F    fd}t          j        |dd          d         S )Nc                 |    t          j        t          | dz
            t          d| z
  dz
            z             S rQ   r  )r}  r

  s    r/   integz'tukeylambda_gen._entropy.<locals>.integ#  s4    6#aQ--AaCQ7888r1   r   r   )r   r  )r:   r

  r
  s    ` r/   r   ztukeylambda_gen._entropy#  s5    	9 	9 	9 	9 	9~eQ**1--r1   N)rv   rw   rx   ry   rV   r^   re   rh   rq   r   r   rz   r1   r/   r
  r
  "  s         ,     M M MN N N" " "8 8 8/ / /. . . . .r1   r
  tukeylambdac                       e Zd Zd ZdS )FitUniformFixedScaleDataErrorc                 $    d|d|df| _         d S )NzInvalid values in `data`.  Maximum likelihood estimation with the uniform distribution and fixed scale requires that data.ptp() <= fscale, but data.ptp() = z and fscale = r~  r  )r:   r  r   s      r/   rA  z&FitUniformFixedScaleDataError.__init__#  s%     
 SS&&&
			r1   N)rv   rw   rx   rA  rz   r1   r/   r
  r
  #  s#        
 
 
 
 
r1   r
  c                   T    e Zd ZdZd ZddZd Zd Zd Zd Z	d	 Z
ed
             ZdS )uniform_gena  A uniform continuous random variable.

    In the standard form, the distribution is uniform on ``[0, 1]``. Using
    the parameters ``loc`` and ``scale``, one obtains the uniform distribution
    on ``[loc, loc + scale]``.

    %(before_notes)s

    %(example)s

    c                     g S rA   rz   r]   s    r/   r^   zuniform_gen._shape_info/#  r   r1   Nc                 0    |                     dd|          S )Nr{   r|   )r  r   s      r/   r   zuniform_gen._rvs2#  s    ##Cd333r1   c                     d||k    z  S r  rz   r   s     r/   re   zuniform_gen._pdf5#  s    AF|r1   c                     |S rA   rz   r   s     r/   rh   zuniform_gen._cdf8#      r1   c                     |S rA   rz   r   s     r/   rq   zuniform_gen._ppf;#  r!
  r1   c                     dS )N)r   gUUUUUU?r   g333333rz   r]   s    r/   r   zuniform_gen._stats>#  s    ##r1   c                     dS r  rz   r]   s    r/   r   zuniform_gen._entropyA#  r  r1   c                 ,   t          |          dk    rt          d          |                    dd          }|                    dd          }t          |           ||t	          d          t          j        |          }t          j        |                                          st	          d          |r|)|	                                }|
                                }n|}|                                |z
  }|	                                |k     rt          d|||z   	          nJ|
                                }||k    rt          ||
          |	                                d||z
  z  z
  }|}t          |          t          |          fS )a	  
        Maximum likelihood estimate for the location and scale parameters.

        `uniform.fit` uses only the following parameters.  Because exact
        formulas are used, the parameters related to optimization that are
        available in the `fit` method of other distributions are ignored
        here.  The only positional argument accepted is `data`.

        Parameters
        ----------
        data : array_like
            Data to use in calculating the maximum likelihood estimate.
        floc : float, optional
            Hold the location parameter fixed to the specified value.
        fscale : float, optional
            Hold the scale parameter fixed to the specified value.

        Returns
        -------
        loc, scale : float
            Maximum likelihood estimates for the location and scale.

        Notes
        -----
        An error is raised if `floc` is given and any values in `data` are
        less than `floc`, or if `fscale` is given and `fscale` is less
        than ``data.max() - data.min()``.  An error is also raised if both
        `floc` and `fscale` are given.

        Examples
        --------
        >>> import numpy as np
        >>> from scipy.stats import uniform

        We'll fit the uniform distribution to `x`:

        >>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])

        For a uniform distribution MLE, the location is the minimum of the
        data, and the scale is the maximum minus the minimum.

        >>> loc, scale = uniform.fit(x)
        >>> loc
        2.0
        >>> scale
        11.0

        If we know the data comes from a uniform distribution where the support
        starts at 0, we can use `floc=0`:

        >>> loc, scale = uniform.fit(x, floc=0)
        >>> loc
        0.0
        >>> scale
        13.0

        Alternatively, if we know the length of the support is 12, we can use
        `fscale=12`:

        >>> loc, scale = uniform.fit(x, fscale=12)
        >>> loc
        1.5
        >>> scale
        12.0

        In that last example, the support interval is [1.5, 13.5].  This
        solution is not unique.  For example, the distribution with ``loc=2``
        and ``scale=12`` has the same likelihood as the one above.  When
        `fscale` is given and it is larger than ``data.max() - data.min()``,
        the parameters returned by the `fit` method center the support over
        the interval ``[data.min(), data.max()]``.

        r   r  r   Nr   r   r   r  r  )r  r   r   )r  r-   r,   r0   r   rC   r   r   r   r  r  r  r<  r
  r  )	r:   r   r;   r.   r   r   r'   r(   r  s	            r/   r9   zuniform_gen.fitD#  s   V t99q==1222xx%%(D))$T*** 2 ) * * * z${4  $$&& 	ECDDD> >|hhjj

 

S(88::##&y3;OOOO $ ((**CV||3FKKKK ((**sFSL11CE Szz5<<''r1   r   )rv   rw   rx   ry   r^   r   re   rh   rq   r   r   r>   r9   rz   r1   r/   r
  r
  ##  s        
 
  4 4 4 4      $ $ $   R( R( _R( R( R(r1   r
  r  c                        e Zd ZdZd ZddZ ee           fd            Zd Z	d Z
d Zd	 Zd
 Z eed          	 	 d fd	            Z xZS )vonmises_gena  A Von Mises continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `vonmises` and `vonmises_line` is:

    .. math::

        f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

    for :math:`-\pi \le x \le \pi`, :math:`\kappa > 0`. :math:`I_0` is the
    modified Bessel function of order zero (`scipy.special.i0`).

    `vonmises` is a circular distribution which does not restrict the
    distribution to a fixed interval. Currently, there is no circular
    distribution framework in scipy. The ``cdf`` is implemented such that
    ``cdf(x + 2*np.pi) == cdf(x) + 1``.

    `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
    on the real line. This is a regular (i.e. non-circular) distribution.

    `vonmises` and `vonmises_line` take ``kappa`` as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS rn  r[   r]   s    r/   r^   zvonmises_gen._shape_info#  rp  r1   Nc                 2    |                     d||          S )Nr{   r  )vonmises)r:   ro  r   r   s       r/   r   zvonmises_gen._rvs#  s    $$S%d$;;;r1   c                      t                      j        |i |}t          j        |t          j        z   dt          j        z            t          j        z
  S r	  )r7   r  rC   modr   )r:   r;   r.   r  r  s       r/   r  zvonmises_gen.rvs$  sB    eggk4(4((vcBEk1RU7++be33r1   c                     t          j        |t          j        |          z            dt           j        z  t          j        |          z  z  S r	  )rC   r   rj   cosm1r   r	  rr  s      r/   re   zvonmises_gen._pdf$  s9    
 veBHQKK'((AbeGBF5MM,ABBr1   c                     |t          j        |          z  t          j        dt          j        z            z
  t          j        t          j        |                    z
  S r	  )rj   r.
  rC   r   r   r	  rr  s      r/   r   zvonmises_gen._logpdf$  s?    rx{{"RVAbeG__4rvbfUmm7L7LLLr1   c                 ,    t          j        ||          S rA   )r   von_mises_cdfrr  s      r/   rh   zvonmises_gen._cdf$  s    #E1---r1   c                     dS r  rz   r  s     r/   _stats_skipzvonmises_gen._stats_skip$  r  r1   c                     | t          j        |          z  t          j        |          z  t          j        dt          j        z  t          j        |          z            z   |z   S r	  )rj   i1er	  rC   r   r   r  s     r/   r   zvonmises_gen._entropy$  sU     &6q25y26%==011249: 	;r1   z        The default limits of integration are endpoints of the interval
        of width ``2*pi`` centered at `loc` (e.g. ``[-pi, pi]`` when
        ``loc=0``).

r   rz   r   r   Fc           	          t           j         t           j        }
}	|||	z   }|||
z   } t                      j        |||||||fi |S rA   )rC   r   r7   r  )r:   rO  r;   r'   r(   lbubconditionalr.   r  r  r  s              r/   r  zvonmises_gen.expect$$  sk     %B:rB:rBuww~dD##R[B B<@B B 	Br1   r   )Nrz   r   r   NNF)rv   rw   rx   ry   r^   r   r
   r   r  re   r   rh   r3
  r   r   r  r  r  s   @r/   r'
  r'
  #  s+        <I I I< < < < M**4 4 4 4 +*4C C CM M M. . .     
; 
; 
; } 5    FJ 
B 
B 
B 
B 
B	 
B 
B 
B 
B 
Br1   r'
  r*
  vonmises_linec                   d    e Zd ZdZej        Zd ZddZd Z	d Z
d Zd Zd	 Zd
 Zd Zd Zd ZdS )r  aX  A Wald continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wald` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })

    for :math:`x >= 0`.

    `wald` is a special case of `invgauss` with ``mu=1``.

    %(after_notes)s

    %(example)s
    c                     g S rA   rz   r]   s    r/   r^   zwald_gen._shape_infoP$  r   r1   Nc                 2    |                     dd|          S r  r  r   s      r/   r   zwald_gen._rvsS$  s      c 555r1   c                 8    t                               |d          S r  )r  re   r   s     r/   re   zwald_gen._pdfV$  s    }}Q$$$r1   c                 8    t                               |d          S r  )r  rh   r   s     r/   rh   zwald_gen._cdfZ$      }}Q$$$r1   c                 8    t                               |d          S r  )r  rl   r   s     r/   rl   zwald_gen._sf]$  s    ||As###r1   c                 8    t                               |d          S r  )r  rq   r   s     r/   rq   zwald_gen._ppf`$  r@
  r1   c                 8    t                               |d          S r  )r  rt   r   s     r/   rt   zwald_gen._isfc$  r@
  r1   c                 8    t                               |d          S r  )r  r   r   s     r/   r   zwald_gen._logpdff$      3'''r1   c                 8    t                               |d          S r  )r  r   r   s     r/   r   zwald_gen._logcdfi$  rE
  r1   c                 8    t                               |d          S r  )r  r   r   s     r/   r   zwald_gen._logsfl$  s    q#&&&r1   c                     dS )N)r|   r|   r  r  rz   r]   s    r/   r   zwald_gen._statso$  s    ""r1   r   )rv   rw   rx   ry   r   r  r  r^   r   re   rh   rl   rq   rt   r   r   r   r   rz   r1   r/   r  r  9$  s         ( "4M  6 6 6 6% % %% % %$ $ $% % %% % %( ( (( ( (' ' '# # # # #r1   r  r  c                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
wrapcauchy_gena  A wrapped Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wrapcauchy` is:

    .. math::

        f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}

    for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.

    `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                     |dk    |dk     z  S r1  rz   r)  s     r/   rV   zwrapcauchy_gen._argcheck$  r  r1   c                 (    t          dddd          gS )Nr  F)r   r   r   r.	  r]   s    r/   r^   zwrapcauchy_gen._shape_info$  s    3v~>>??r1   c                 z    d||z  z
  dt           j        z  d||z  z   d|z  t          j        |          z  z
  z  z  S rS  r  r  s      r/   re   zwrapcauchy_gen._pdf$  s=    AaC!BE'1QqS51RVAYY#6788r1   c                 j    d }d }d|z   d|z
  z  }t          |t          j        k     ||f||          S )Nc                 z    dt           j        z  t          j        |t          j        | dz            z            z  S r"  rC   r   r4  r8  rd   crs     r/   r  zwrapcauchy_gen._cdf.<locals>.f1$  s-    RU7RYr"&1++~6666r1   c           	          ddt           j        z  t          j        |t          j        dt           j        z  | z
  dz            z            z  z
  S r"  rP
  rQ
  s     r/   r  zwrapcauchy_gen._cdf.<locals>.f2$  s?    qw2bfagk1_.E.E+E!F!FFFFr1   r   r  )r   rC   r   )r:   rd   r  r  r  rR
  s         r/   rh   zwrapcauchy_gen._cdf$  sW    	7 	7 	7	G 	G 	G !ea!e_!be)aWr::::r1   c           
      V   d|z
  d|z   z  }dt          j        |t          j        t           j        |z            z            z  }dt           j        z  dt          j        |t          j        t           j        d|z
  z            z            z  z
  }t          j        |dk     ||          S )Nr|   rH   r   r   )rC   r4  r8  r   r  )r:   rp   r  r  rcqrcmqs         r/   rq   zwrapcauchy_gen._ppf$  s    1us1uo	#bfRU1Woo-...wq3rvbeQqSk':':#:;;;;xE	3---r1   c                 V    t          j        dt           j        z  d||z  z
  z            S r  r   r)  s     r/   r   zwrapcauchy_gen._entropy$  s$    vagq1uo&&&r1   c                 t    dt          j        |          t          j        |          dt           j        z  z  fS rZ  )rC   r  r  r   )r:   r   s     r/   r  zwrapcauchy_gen._fitstart$  s,     BF4LL"&,,"%"888r1   N)rv   rw   rx   ry   rV   r^   re   rh   rq   r   r  rz   r1   r/   rJ
  rJ
  v$  s         *! ! !@ @ @9 9 9; ; ;. . .' ' '9 9 9 9 9r1   rJ
  
wrapcauchyc                   P    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
 ZddZdS )gennorm_gena0  A generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution
    norm : normal distribution

    Notes
    -----
    The probability density function for `gennorm` is [1]_:

    .. math::

        f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),

    where :math:`x` is a real number, :math:`\beta > 0` and
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to a Laplace distribution.
    For :math:`\beta = 2`, it is identical to a normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    .. [2] Nardon, Martina, and Paolo Pianca. "Simulation techniques for
           generalized Gaussian densities." Journal of Statistical
           Computation and Simulation 79.11 (2009): 1317-1329

    .. [3] Wicklin, Rick. "Simulate data from a generalized Gaussian
           distribution" in The DO Loop blog, September 21, 2016,
           https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html

    %(example)s

    c                 @    t          dddt          j        fd          gS Nr_  Fr   r   r[   r]   s    r/   r^   zgennorm_gen._shape_info$      651bf+~FFGGr1   c                 R    t          j        |                     ||                    S rA   r  r:   rd   r_  s      r/   re   zgennorm_gen._pdf$  s     vdll1d++,,,r1   c                     t          j        d|z            t          j        d|z            z
  t	          |          |z  z
  S r   )rC   r   rj   rW  r  r`
  s      r/   r   zgennorm_gen._logpdf$  s8    vc$h"*SX"6"66QEEr1   c                     dt          j        |          z  }d|z   |t          j        d|z  t	          |          |z            z  z
  S r   )rC   rD   rj   r_  r  r:   rd   r_  r  s       r/   rh   zgennorm_gen._cdf$  sB    "'!**a1r|CHc!ffdlCCCCCr1   c                     t          j        |dz
            }|t          j        d|z  d|z   d|z  |z  z
            d|z  z  z  S )Nr   r|   r   )rC   rD   rj   rg  rc
  s       r/   rq   zgennorm_gen._ppf$  sJ    GAG2?3t8cAgQq-@AACHMMMr1   c                 0    |                      | |          S rA   rb  r`
  s      r/   rl   zgennorm_gen._sf$  s    yy!T"""r1   c                 0    |                      ||           S rA   rf  r`
  s      r/   rt   zgennorm_gen._isf$  s    		!T""""r1   c                     t          j        d|z  d|z  d|z  g          \  }}}dt          j        ||z
            dt          j        ||z   d|z  z
            dz
  fS )Nr|   r  r  r{   r   )rj   rW  rC   r   )r:   r_  c1c3c5s        r/   r   zgennorm_gen._stats$  sa    ZT3t8SX >??
B26"r'??BrBwR/?(@(@2(EEEr1   c                 l    d|z  t          j        d|z            z
  t          j        d|z            z   S r  ru  r:   r_  s     r/   r   zgennorm_gen._entropy$  s2    Dy26"t),,,rz"t)/D/DDDr1   Nc                     |                     d|z  |          }|d|z  z  }t          j        |          }|                    |j                  dk     }||          ||<   |S )Nr   r  r   )r  rC   r   randomr  )r:   r_  r   r   r"  rV  rL  s          r/   r   zgennorm_gen._rvs%  sk     qvD11!D&MJqMM"""0036T7($r1   r   )rv   rw   rx   ry   r^   re   r   rh   rq   rl   rt   r   r   r   rz   r1   r/   r[
  r[
  $  s        ) )TH H H- - -F F FD D D
N N N
# # ## # #F F FE E E	 	 	 	 	 	r1   r[
  gennormc                   B    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	 Zd
S )halfgennorm_gena  The upper half of a generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    gennorm : generalized normal distribution
    expon : exponential distribution
    halfnorm : half normal distribution

    Notes
    -----
    The probability density function for `halfgennorm` is:

    .. math::

        f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)

    for :math:`x, \beta > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `halfgennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to an exponential distribution.
    For :math:`\beta = 2`, it is identical to a half normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    %(example)s

    c                 @    t          dddt          j        fd          gS r]
  r[   r]   s    r/   r^   zhalfgennorm_gen._shape_info4%  r^
  r1   c                 R    t          j        |                     ||                    S rA   r  r`
  s      r/   re   zhalfgennorm_gen._pdf7%  s"     vdll1d++,,,r1   c                 f    t          j        |          t          j        d|z            z
  ||z  z
  S r  ru  r`
  s      r/   r   zhalfgennorm_gen._logpdf=%  s,    vd||bjT222QW<<r1   c                 8    t          j        d|z  ||z            S r  r[  r`
  s      r/   rh   zhalfgennorm_gen._cdf@%  s    {3t8QW---r1   c                 >    t          j        d|z  |          d|z  z  S r  r  r`
  s      r/   rq   zhalfgennorm_gen._ppfC%  s!    ~c$h**SX66r1   c                 8    t          j        d|z  ||z            S r  r^  r`
  s      r/   rl   zhalfgennorm_gen._sfF%  s    |CHag...r1   c                 >    t          j        d|z  |          d|z  z  S r  rn  r`
  s      r/   rt   zhalfgennorm_gen._isfI%  s!    s4x++c$h77r1   c                 f    d|z  t          j        |          z
  t          j        d|z            z   S r  ru  rl
  s     r/   r   zhalfgennorm_gen._entropyL%  s,    4x"&,,&CH)=)===r1   N)rv   rw   rx   ry   r^   re   r   rh   rq   rl   rt   r   rz   r1   r/   rq
  rq
  %  s        " "FH H H- - -= = =. . .7 7 7/ / /8 8 8> > > > >r1   rq
  halfgennormc                   L     e Zd ZdZd Zd Z fdZd Zd Zd Z	d Z
d	 Z xZS )
crystalball_gena  
    Crystalball distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `crystalball` is:

    .. math::

        f(x, \beta, m) =  \begin{cases}
                            N \exp(-x^2 / 2),  &\text{for } x > -\beta\\
                            N A (B - x)^{-m}  &\text{for } x \le -\beta
                          \end{cases}

    where :math:`A = (m / |\beta|)^m  \exp(-\beta^2 / 2)`,
    :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.

    `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
    parameters.  :math:`\beta` defines the point where the pdf changes
    from a power-law to a Gaussian distribution.  :math:`m` is the power
    of the power-law tail.

    References
    ----------
    .. [1] "Crystal Ball Function",
           https://en.wikipedia.org/wiki/Crystal_Ball_function

    %(after_notes)s

    .. versionadded:: 0.19.0

    %(example)s
    c                     |dk    |dk    z  S )z@
        Shape parameter bounds are m > 1 and beta > 0.
        r   r   rz   )r:   r_  rT  s      r/   rV   zcrystalball_gen._argcheckw%  s     A$(##r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )Nr_  Fr   r   rT  r   r[   )r:   ibetaims      r/   r^   zcrystalball_gen._shape_info}%  s>    651bf+~FFUQK@@r{r1   c                 J    t                                          |d          S )N)r   ri  r  r  r  s     r/   r  zcrystalball_gen._fitstart%  r  r1   c                     d||z  |dz
  z  t          j        |dz   dz            z  t          t          |          z  z   z  }d }d }|t	          || k    |||f||          z  S )a`  
        Return PDF of the crystalball function.

                                            --
                                           | exp(-x**2 / 2),  for x > -beta
        crystalball.pdf(x, beta, m) =  N * |
                                           | A * (B - x)**(-m), for x <= -beta
                                            --
        r|   r   rH   r   c                 8    t          j        | dz   dz            S r	  r  rd   r_  rT  s      r/   rhsz!crystalball_gen._pdf.<locals>.rhs%  s    61a4%!)$$$r1   c                 j    ||z  |z  t          j        |dz   dz            z  ||z  |z
  | z
  | z  z  S r   r  r
  s      r/   lhsz!crystalball_gen._pdf.<locals>.lhs%  sG    tVaK"&$'C"8"88tVd]Q&1"-. /r1   r  rC   r   r   r   r   r:   rd   r_  rT  r  r
  r
  s          r/   re   zcrystalball_gen._pdf%  s     1T6QqS>BFD!G8c>$:$::401 2	% 	% 	%	/ 	/ 	/ :a4%i!T1EEEEEr1   c                     d||z  |dz
  z  t          j        |dz   dz            z  t          t          |          z  z   z  }d }d }t          j        |          t          || k    |||f||          z   S )zH
        Return the log of the PDF of the crystalball function.
        r|   r   rH   r   c                     | dz   dz  S r	  rz   r
  s      r/   r
  z$crystalball_gen._logpdf.<locals>.rhs%  s    qD57Nr1   c                     |t          j        ||z            z  |dz  dz  z
  |t          j        ||z  |z
  | z
            z  z
  S r	  r#  r
  s      r/   r
  z$crystalball_gen._logpdf.<locals>.lhs%  sG    RVAdF^^#dAgai/!BF1T6D=1;L4M4M2MMMr1   r  )rC   r   r   r   r   r   r
  s          r/   r   zcrystalball_gen._logpdf%  s     1T6QqS>BFD!G8c>$:$::401 2	 	 		N 	N 	N vayy:a4%i!T1MMMMMr1   c                     d||z  |dz
  z  t          j        |dz   dz            z  t          t          |          z  z   z  }d }d }|t	          || k    |||f||          z  S )z8
        Return CDF of the crystalball function
        r|   r   rH   r   c                     ||z  t          j        |dz   dz            z  |dz
  z  t          t          |           t          |           z
  z  z   S NrH   r   r   rC   r   r   r   r
  s      r/   r
  z!crystalball_gen._cdf.<locals>.rhs%  sT    tVrvtQwhn5551=9Q<<)TE2B2B#BCD Er1   c                 |    ||z  |z  t          j        |dz   dz            z  ||z  |z
  | z
  | dz   z  z  |dz
  z  S r
  r  r
  s      r/   r
  z!crystalball_gen._cdf.<locals>.lhs%  sW    tVaK"&$'C"8"88tVd]Q&1"Q$/034Q38 9r1   r  r
  r
  s          r/   rh   zcrystalball_gen._cdf%  s     1T6QqS>BFD!G8c>$:$::401 2	E 	E 	E	9 	9 	9 :a4%i!T1EEEEEr1   c                    d||z  |dz
  z  t          j        |dz   dz            z  t          t          |          z  z   z  }|||z  z  t          j        |dz   dz            z  |dz
  z  }d }d }t	          ||k     |||f||          S )Nr|   r   rH   r   c                     t          j        |dz   dz            }||z  |z  |dz
  z  }d|t          t          |          z  z   z  }||z  |z
  |dz
  ||z  | z  z  |z  | z  |z  dd|z
  z  z  z
  S r  r
  r}  r_  rT  eb2Cr  s         r/   ppf_lessz&crystalball_gen._ppf.<locals>.ppf_less%  s    &$'!$$C43!A#&A1{Yt__445AdFTM!eaf^+C/1!3q!A#w?@ Ar1   c                     t          j        |dz   dz            }||z  |z  |dz
  z  }d|t          t          |          z  z   z  }t	          t          |           dt          z  | |z  |z
  z  z             S r  )rC   r   r   r   r   r
  s         r/   ppf_greaterz)crystalball_gen._ppf.<locals>.ppf_greater%  sy    &$'!$$C43!A#&A1{Yt__445AYu--;1q0IIJJJr1   r  r
  )r:   r}  r_  rT  r  pbetar
  r
  s           r/   rq   zcrystalball_gen._ppf%  s    1T6QqS>BFD!G8c>$:$::401 2QtVrvtQwhqj111QU;	A 	A 	A	K 	K 	K !e)aq\X+NNNNr1   c           	         d||z  |dz
  z  t          j        |dz   dz            z  t          t          |          z  z   z  }d }|t	          |dz   |k     |||ft          j        |t           j        g          t           j                  z  S )zR
        Returns the n-th non-central moment of the crystalball function.
        r|   r   rH   r   c                    ||z  |z  t          j        |dz   dz            z  }||z  |z
  }d| dz
  dz  z  t          j        | dz   dz            z  dd| z  t          j        | dz   dz  |dz  dz            z  z   z  }t          j        |j                  }t          | dz             D ]B}|t          j        | |          || |z
  z  z  d|z  z  ||z
  dz
  z  ||z  | |z   dz   z  z  z  }C||z  |z   S )z
            Returns n-th moment. Defined only if n+1 < m
            Function cannot broadcast due to the loop over n
            rH   r   r   r|   r  )	rC   r   rj   r  r\  r  r  r  binom)rU   r_  rT  r  Br
  r
  r  s           r/   r	  z*crystalball_gen._munp.<locals>.n_th_moment%  s    
 4!bfdAgX^444A$A!Sy>BHac1W$5$552'BK1aq1$E$EEEGC(39%%C1q5\\ 0 0AQqS1R!G;q1uqyI4A26A:./ 0s7S= r1   r	  )rC   r   r   r   r   r  r	  r\   )r:   rU   r_  rT  r  r	  s         r/   r   zcrystalball_gen._munp%  s     1T6QqS>BFD!G8c>$:$::401 2	! 	! 	! :a!eai!T1 l;
|LLL f& & & 	&r1   )rv   rw   rx   ry   rV   r^   r  re   r   rh   rq   r   r  r  s   @r/   r|
  r|
  S%  s        " "F$ $ $  
6 6 6 6 6F F F,N N NF F F"O O O(& & & & & & &r1   r|
  crystalballzA Crystalball Function)r   longnamec                 >    t          j        d| dz  dz            dz  S )a  
    Utility function for the argus distribution used in the pdf, sf and
    moment calculation.
    Note that for all x > 0:
    gammainc(1.5, x**2/2) = 2 * (_norm_cdf(x) - x * _norm_pdf(x) - 0.5).
    This can be verified directly by noting that the cdf of Gamma(1.5) can
    be written as erf(sqrt(x)) - 2*sqrt(x)*exp(-x)/sqrt(Pi).
    We use gammainc instead of the usual definition because it is more precise
    for small chi.
    ri  rH   r[  )rl  s    r/   
_argus_phir
  %  s#     ;sCF1H%%))r1   c                   F    e Zd ZdZd Zd Zd Zd Zd ZddZ	dd	Z
d
 ZdS )	argus_gena  
    Argus distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `argus` is:

    .. math::

        f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
                     \exp(-\chi^2 (1 - x^2)/2)

    for :math:`0 < x < 1` and :math:`\chi > 0`, where

    .. math::

        \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2

    with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
    normal distribution, respectively.

    `argus` takes :math:`\chi` as shape a parameter.

    %(after_notes)s

    References
    ----------
    .. [1] "ARGUS distribution",
           https://en.wikipedia.org/wiki/ARGUS_distribution

    .. versionadded:: 0.19.0

    %(example)s
    c                 @    t          dddt          j        fd          gS )Nrl  Fr   r   r[   r]   s    r/   r^   zargus_gen._shape_info!&  s    5%!RVnEEFFr1   c                 p   t          j        d          5  d||z  z
  }dt          j        |          z  t          z
  t          j        t	          |                    z
  }|t          j        |          z   dt          j        | |z            z  z   |dz  |z  dz  z
  cd d d            S # 1 swxY w Y   d S )Nr+  r,  r|   r  r   rH   )rC   r.  r   r   r
  r  )r:   rd   rl  rV  r  s        r/   r   zargus_gen._logpdf$&  s    [))) 	G 	Gac	A"&++.
31H1HHArvayy=3rx1~~#55Q
QF	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	Gs   BB++B/2B/c                 R    t          j        |                     ||                    S rA   r  r:   rd   rl  s      r/   re   zargus_gen._pdf+&  s     vdll1c**+++r1   c                 4    d|                      ||          z
  S r  r$  r
  s      r/   rh   zargus_gen._cdf.&  s    TXXa%%%%r1   c                 v    t          |t          j        d|dz  z
            z            t          |          z  S r"  )r
  rC   r   r
  s      r/   rl   zargus_gen._sf1&  s2    #AqD 1 1122Z__DDr1   Nc                   	
 t          j        |          }|j        dk    r|                     |||          }nt	          |j        |          \  }	t          t          j        |                    }t          j        |          }t          j	        |gdgdgg          

j
        st          	
fdt          t          |           d          D                       }|                     
d         ||          }|                    |          ||<   
                                 
j
        |dk    r|d         }|S )	Nr   )r  r   r  r  r  c              3   `   K   | ](}|         sj         |         nt          d           V  )d S rA   r  r  s     r/   r  z!argus_gen._rvs.<locals>.<genexpr>A&  r  r1   r   rz   )rC   r   r   r  r   r  r  r  r  r  r  r  r  r  r  r  )r:   rl  r   r   r  r  r  r  r  r  r  s            @@r/   r   zargus_gen._rvs4&  sb   joo8q==""340< # > >CC #39d33GCRWS\\**J(4..CC5"/&0\N4 4 4B k  ; ; ; ; ;%*CII:q%9%9; ; ; ; ;$$RUz2> % @ @99S>>C k  2::b'C
r1   c                    t          t          j        |                    }t          t          j        |                    }t          j        |          }d}||z  }|dk    r| dz  }	||k     r||z
  }
|                    |
          }|                    |
          }|dz  }t          j        |          |	|z  k    }t          j        |          }|dk    r,t          j	        d||         z
            }|||||z   <   ||z  }||k     nN|dk    rt          j
        | dz            }||k     r||z
  }
|                    |
          }|                    |
          }dt          j        |d|z
  z  |z             z  |z  }|dz  |z   dk    }t          j        |          }|dk    r,t          j	        d||         z             }|||||z   <   ||z  }||k     n}||k     rZ||z
  }
|                    d|
          }||dz  k    }t          j        |          }|dk    r||         ||||z   <   ||z  }||k     Zt          j	        dd|z  |z  z
            }t          j        ||          S )	Nr   r   rH   r  gUUUUUU?r   g?ri  )r  rC   r  r  r  r  r  r   r  r   r   rg  r  )r:   rl  r  r   r  r  rd   r  rR  r  r  r  r  r"  r  r  r  echirD  s                      r/   r  zargus_gen._rvs_scalarL&  s   h r}Z0011  HQKK	Sy#::	Aa--	M ((a(00 ((a(00H&))q1u,VF^^
>>'!ai-00C<?AiZ!789+I a-- CZZ64%!)$$Da--	M ((a(00 ((a(00tq1u~1222T9 Q$(a-VF^^
>>'!ai-00C<?AiZ!789+I a-- a--	M //!/<<tax-VF^^
>><=fIAiZ!789+I a-- AEDL())Az!V$$$r1   c                    t          j        |t                    }t          |          }t          j        t           j        dz            |z  t          j        d|dz  dz            z  |z  }t          j        |          }|dk    }||         }dd|dz  z  z
  |t          |          z  ||         z  z   ||<   ||          }g d}t          j
        ||          || <   |||dz  z
  d d fS )	Nr
  r  r   rH   r  g?r  )	g_1g־r   gWBar   gp|RH?r   gE'卡?r   g?)rC   r   r  r
  r   r   rj   r  r	  r   r  )r:   rl  r  rT  r6  rL  r  coefs           r/   r   zargus_gen._stats&  s     jE***ooGBE!Gs"RVAsAvax%8%883>mC  SyIAqDL1y||#3c$i#??D	JKKKZa((TE
#1*dD((r1   r   )rv   rw   rx   ry   r^   r   re   rh   rl   r   r  r   rz   r1   r/   r
  r
  %  s        # #HG G GG G G, , ,& & &E E E   0c% c% c% c%J) ) ) ) )r1   r
  arguszAn Argus Function)r   r
  r~   r   c                   ^     e Zd ZdZej        Zdd fd
Zd Zd Zd Z	d Z
d	 Z fd
Z xZS )rv_histograma  
    Generates a distribution given by a histogram.
    This is useful to generate a template distribution from a binned
    datasample.

    As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
    a collection of generic methods (see `rv_continuous` for the full list),
    and implements them based on the properties of the provided binned
    datasample.

    Parameters
    ----------
    histogram : tuple of array_like
        Tuple containing two array_like objects.
        The first containing the content of n bins,
        the second containing the (n+1) bin boundaries.
        In particular, the return value of `numpy.histogram` is accepted.

    density : bool, optional
        If False, assumes the histogram is proportional to counts per bin;
        otherwise, assumes it is proportional to a density.
        For constant bin widths, these are equivalent, but the distinction
        is important when bin widths vary (see Notes).
        If None (default), sets ``density=True`` for backwards compatibility,
        but warns if the bin widths are variable. Set `density` explicitly
        to silence the warning.

        .. versionadded:: 1.10.0

    Notes
    -----
    When a histogram has unequal bin widths, there is a distinction between
    histograms that are proportional to counts per bin and histograms that are
    proportional to probability density over a bin. If `numpy.histogram` is
    called with its default ``density=False``, the resulting histogram is the
    number of counts per bin, so ``density=False`` should be passed to
    `rv_histogram`. If `numpy.histogram` is called with ``density=True``, the
    resulting histogram is in terms of probability density, so ``density=True``
    should be passed to `rv_histogram`. To avoid warnings, always pass
    ``density`` explicitly when the input histogram has unequal bin widths.

    There are no additional shape parameters except for the loc and scale.
    The pdf is defined as a stepwise function from the provided histogram.
    The cdf is a linear interpolation of the pdf.

    .. versionadded:: 0.19.0

    Examples
    --------

    Create a scipy.stats distribution from a numpy histogram

    >>> import scipy.stats
    >>> import numpy as np
    >>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5, random_state=123)
    >>> hist = np.histogram(data, bins=100)
    >>> hist_dist = scipy.stats.rv_histogram(hist, density=False)

    Behaves like an ordinary scipy rv_continuous distribution

    >>> hist_dist.pdf(1.0)
    0.20538577847618705
    >>> hist_dist.cdf(2.0)
    0.90818568543056499

    PDF is zero above (below) the highest (lowest) bin of the histogram,
    defined by the max (min) of the original dataset

    >>> hist_dist.pdf(np.max(data))
    0.0
    >>> hist_dist.cdf(np.max(data))
    1.0
    >>> hist_dist.pdf(np.min(data))
    7.7591907244498314e-05
    >>> hist_dist.cdf(np.min(data))
    0.0

    PDF and CDF follow the histogram

    >>> import matplotlib.pyplot as plt
    >>> X = np.linspace(-5.0, 5.0, 100)
    >>> fig, ax = plt.subplots()
    >>> ax.set_title("PDF from Template")
    >>> ax.hist(data, density=True, bins=100)
    >>> ax.plot(X, hist_dist.pdf(X), label='PDF')
    >>> ax.plot(X, hist_dist.cdf(X), label='CDF')
    >>> ax.legend()
    >>> fig.show()

    N)densityc                8   || _         || _        t          |          dk    rt          d          t	          j        |d                   | _        t	          j        |d                   | _        t          | j                  dz   t          | j                  k    rt          d          | j        dd         | j        dd         z
  | _        t	          j	        | j        | j        d                    }|#|r!d}t          j        |t          d	           d
}n|s| j        | j        z  | _        | j        t          t	          j        | j        | j        z                      z  | _        t	          j        | j        | j        z            | _        t	          j        d| j        dg          | _        t	          j        d| j        g          | _        | j        d         x|d<   | _        | j        d         x|d<   | _         t)                      j        |i | dS )a5  
        Create a new distribution using the given histogram

        Parameters
        ----------
        histogram : tuple of array_like
            Tuple containing two array_like objects.
            The first containing the content of n bins,
            the second containing the (n+1) bin boundaries.
            In particular, the return value of np.histogram is accepted.
        density : bool, optional
            If False, assumes the histogram is proportional to counts per bin;
            otherwise, assumes it is proportional to a density.
            For constant bin widths, these are equivalent.
            If None (default), sets ``density=True`` for backward
            compatibility, but warns if the bin widths are variable. Set
            `density` explicitly to silence the warning.
        rH   z)Expected length 2 for parameter histogramr   r   zbNumber of elements in histogram content and histogram boundaries do not match, expected n and n+1.Nr  zjBin widths are not constant. Assuming `density=True`.Specify `density` explicitly to silence this warning.r  Tr{   r~   r   )
_histogram_densityr  r   rC   r   _hpdf_hbins_hbin_widthsallcloserp  r  r  r  r  cumsum_hcdfhstackr~   r   r7   rA  )r:   	histogramr
  r;   kwargs	bins_varyro  r  s          r/   rA  zrv_histogram.__init__"'  s   & $y>>QHIIIZ	!--
j1..tz??Q#dk"2"222 3 4 4 4 !KOdk#2#.>>D$5t7H7KLLL	?y?OGM'>a@@@@GG 	8d&77DJZ%tzD<M/M(N(N"O"OO
YtzD,==>>
YTZ566
YTZ011
#{1~-sdf#{2.sdf$)&)))))r1   c                 P    | j         t          j        | j        |d                   S )z&
        PDF of the histogram
        r*  )side)r
  rC   searchsortedr
  r   s     r/   re   zrv_histogram._pdfR'  s$     z"/$+qwGGGHHr1   c                 B    t          j        || j        | j                  S )z3
        CDF calculated from the histogram
        )rC   interpr
  r
  r   s     r/   rh   zrv_histogram._cdfX'  s     yDK444r1   c                 B    t          j        || j        | j                  S )zC
        Percentile function calculated from the histogram
        )rC   r
  r
  r
  r   s     r/   rq   zrv_histogram._ppf^'  s     yDJ444r1   c                     | j         dd         |dz   z  | j         dd         |dz   z  z
  |dz   z  }t          j        | j        dd         |z            S )z$Compute the n-th non-central moment.r   Nr  )r
  rC   r  r
  )r:   rU   	integralss      r/   r   zrv_histogram._munpd'  s\    [_qs+dk#2#.>1.EE!A#N	vdj2&2333r1   c                     t          | j        dd         dk    | j        dd         ft          j        d          }t          j        | j        dd         |z  | j        z             S )zCompute entropy of distributionr   r  r{   )r   r
  rC   r   r  r
  )r:   r  s     r/   r   zrv_histogram._entropyi'  si    AbD)C/*QrT*,  tz!B$'#-0AABBBBr1   c                 p    t                                                      }| j        |d<   | j        |d<   |S )zF
        Set the histogram as additional constructor argument
        r
  r
  )r7   _updated_ctor_paramr
  r
  )r:   dctr  s     r/   r
  z rv_histogram._updated_ctor_paramq'  s6     gg))++?KI
r1   )rv   rw   rx   ry   r   r  rA  re   rh   rq   r   r   r
  r  r  s   @r/   r
  r
  &  s        Y Yt "/M15 .* .* .* .* .* .* .*`I I I5 5 55 5 54 4 4
C C C        r1   r
  c                   @     e Zd ZdZd Zd Z fdZd Zd Zd Z	 xZ
S )studentized_range_genu  A studentized range continuous random variable.

    %(before_notes)s

    See Also
    --------
    t: Student's t distribution

    Notes
    -----
    The probability density function for `studentized_range` is:

    .. math::

         f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
                        2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
                        s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
                        [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds

    for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`.

    `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
    as shape parameters.

    When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
    degrees of freedom) is used to compute the cumulative distribution
    function [4]_ and probability distribution function.

    %(after_notes)s

    References
    ----------

    .. [1] "Studentized range distribution",
           https://en.wikipedia.org/wiki/Studentized_range_distribution
    .. [2] Batista, Ben Dêivide, et al. "Externally Studentized Normal Midrange
           Distribution." Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp.
           378-389., doi:10.1590/1413-70542017414047716.
    .. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals
           of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147.
           JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021.
    .. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and
           Upper Quantiles for the Studentized Range." Journal of the Royal
           Statistical Society. Series C (Applied Statistics), vol. 32, no. 2,
           1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18
           Feb. 2021.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import studentized_range
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> k, df = 3, 10
    >>> mean, var, skew, kurt = studentized_range.stats(k, df, moments='mvsk')

    Display the probability density function (``pdf``):

    >>> x = np.linspace(studentized_range.ppf(0.01, k, df),
    ...                 studentized_range.ppf(0.99, k, df), 100)
    >>> ax.plot(x, studentized_range.pdf(x, k, df),
    ...         'r-', lw=5, alpha=0.6, label='studentized_range pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = studentized_range(k, df)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df)
    >>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df))
    True

    Rather than using (``studentized_range.rvs``) to generate random variates,
    which is very slow for this distribution, we can approximate the inverse
    CDF using an interpolator, and then perform inverse transform sampling
    with this approximate inverse CDF.

    This distribution has an infinite but thin right tail, so we focus our
    attention on the leftmost 99.9 percent.

    >>> a, b = studentized_range.ppf([0, .999], k, df)
    >>> a, b
    0, 7.41058083802274

    >>> from scipy.interpolate import interp1d
    >>> rng = np.random.default_rng()
    >>> xs = np.linspace(a, b, 50)
    >>> cdf = studentized_range.cdf(xs, k, df)
    # Create an interpolant of the inverse CDF
    >>> ppf = interp1d(cdf, xs, fill_value='extrapolate')
    # Perform inverse transform sampling using the interpolant
    >>> r = ppf(rng.uniform(size=1000))

    And compare the histogram:

    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                     |dk    |dk    z  S r  rz   )r:   r  rO  s      r/   rV   zstudentized_range_gen._argcheck'  s    A"q&!!r1   c                     t          dddt          j        fd          }t          dddt          j        fd          }||gS )Nr  Fr   r   rO  r   r[   )r:   r5  r  s      r/   r^   z!studentized_range_gen._shape_info'  s>    UQK@@uq"&k>BBCyr1   c                 X    t          t          |                               |d          S )N)rH   r   r  )r7   r
  r  r  s     r/   r  zstudentized_range_gen._fitstart'  s'    *D11;;Dv;NNNr1   c                     d|                                  \  fd}t          j        |dd          }t          j         ||||                    S )N_studentized_range_momentc                    t          j        ||          }| |||g}t          j        |t                    j                            t
          j                  }t          j	        t           |          }t          j
         t          j
        fdt          j
        f	
fg}t          dd          }t          j        |||          d         S )Nr   rZ  -q=epsabsepsrelrangesopts)r   _studentized_range_pdf_logconstrC   r  r  r  r  r  r   r  r\   dictr   nquad)r  r  rO  	log_constargusr_datar  r
  r
  r  r  cython_symbols            r/   _single_momentz3studentized_range_gen._munp.<locals>._single_moment'  s    >q"EEIaY'CxU++2::6?KKH".v}hOOCw'!RVr2h?FuU333D?3vDAAA!DDr1   r  r   )r   rC   
frompyfuncr	  )	r:   r  r  rO  r
  ufuncr  r  r
  s	         @@@r/   r   zstudentized_range_gen._munp'  sz    3""$$B
	E 
	E 
	E 
	E 
	E 
	E 
	E na33z%%1b//***r1   c                 r    d }t          j        |dd          }t          j         ||||                    S )Nc                 Z   |dk     rd}t          j        ||          }| |||g}t          j        |t                    j                            t
          j                  }t          j         t          j        fdt          j        fg}n\d}| |g}t          j        |t                    j                            t
          j                  }t          j         t          j        fg}t          j
        t           ||          }t          dd          }	t          j        |||	          d         S )	N順 _studentized_range_pdfr   !_studentized_range_pdf_asymptoticrZ  r
  r
  r
  )r   r
  rC   r  r  r  r  r  r\   r   r  r
  r   r
  
rp   r  rO  r
  r
  r
  r
  r
  r  r
  s
             r/   _single_pdfz/studentized_range_gen._pdf.<locals>._single_pdf(  s     F{{ 8"B1bII	!R+8C//6>>vOOF7BF+a[9 !D!f8C//6>>vOOF7BF+,".v}hOOCuU333D?3vDAAA!DDr1   r  r   )rC   r
  r	  )r:   rd   r  rO  r
  r
  s         r/   re   zstudentized_range_gen._pdf(  sE    	E 	E 	E( k1a00z%%1b//***r1   c           	          d }t          j        |dd          }t          j        t          j         ||||                    dd          S )Nc                 Z   |dk     rd}t          j        ||          }| |||g}t          j        |t                    j                            t
          j                  }t          j         t          j        fdt          j        fg}n\d}| |g}t          j        |t                    j                            t
          j                  }t          j         t          j        fg}t          j
        t           ||          }t          dd          }	t          j        |||	          d         S )	Nr
  _studentized_range_cdfr   !_studentized_range_cdf_asymptoticrZ  r
  r
  r
  )r   _studentized_range_cdf_logconstrC   r  r  r  r  r  r\   r   r  r
  r   r
  r
  s
             r/   _single_cdfz/studentized_range_gen._cdf.<locals>._single_cdf&(  s    
 F{{ 8"B1bII	!R+8C//6>>vOOF7BF+a[9 !D!f8C//6>>vOOF7BF+,".v}hOOCuU333D?3vDAAA!DDr1   r  r   r   )rC   r
  r,  r	  )r:   rd   r  rO  r
  r
  s         r/   rh   zstudentized_range_gen._cdf$(  sU    	E 	E 	E, k1a00 wrz%%1b//22Aq999r1   )rv   rw   rx   ry   rV   r^   r  r   re   rh   r  r  s   @r/   r
  r
  {'  s        l l\" " "  
O O O O O+ + +*+ + +2: : : : : : :r1   r
  studentized_range)r   r~   r   rA   (A  rp  collections.abcr   	functoolsr   r   r  numpyrC   numpy.polynomialr   scipy._lib.doccerr   r	   r
   scipy._lib._ccallbackr   scipyr   r   scipy.specialspecialrj   scipy.special._ufuncs_ufuncsra   scipy._lib._utilr   r   rF  r   _tukeylambda_statsr   r
  r   r
  _distn_infrastructurer   r   r   r   r   r   r   _ksstatsr   r   r   
_constantsr   r   r    r!   r"   r#   scipy.stats._boostr  ra  scipy.optimizer$   scipy.stats._warnings_errorsr%   scipy.statsr0   r>   rM   rO   r}   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r&  r(  r:  r   r<  rJ   rP  rT  rV  r_  r  r  r  r  r  r  r  r  r  r,  r.  rJ  rL  rl  rn  rR  r  r  r  r  r  r  r  r  r  r  r  r  r
  r  r  r6  r8  rG  rI  r  ri  rt  rv  r  r  r  r  r  r  r  r  r	  r  r  r  rQ  ra  rc  r  r|  r  r  r  r  r  r  r  r  r  r  r  r  r!  r2  r4  r=  r?  rJ  rL  r[  r]  re  rg  ro  rq  r  r  r  r  r  r  r3  r5  rB  rD  rO  rQ  rX  rZ  r]  rl  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  _gibrat_method_namesrT  	deprecategetattrr=   setattrr#  r1  r3  rA  rC  r  r  r  r  r  r  r  r  r  r  r  r  r   r$  r'  r@  rB  rQ  re  ro  rq  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r 	  r	  r	  r	  r	  r)	  r+	  r5	  r7	  r^	  r`	  r}	  r~	  ry   r	  r	  r	  r	  r	  r	  r	  r	  r	  r	  r
  r
  r
  r
  r
  r  r'
  r*
  r:
  r  r  rJ
  rY
  r[
  ro
  rq
  rz
  r|
  r
  r
  r
  r
  r
  r
  r\   r
  listglobalsrs  itemspairs_distn_names_distn_gen_names__all__rz   r1   r/   <module>r     s7"    $ $ $ $ $ $ , , , , , , , ,      ' ' ' ' ' '7 7 7 7 7 7 7 7 7 7 3 2 2 2 2 2                   # # # # # # # # # 4 4 4 4 4 4 4 4      B B B B B B B BJ J J J J J J J J J J J J J J J J J 2 1 1 1 1 1 1 1 1 1? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? # # # # # # # # # & & & & & & 1 1 1 1 1 1      9 9 9&
 
 
   .9! 9! 9! 9! 9! 9! 9! 9!x 		C3W---<) <) <) <) <) <) <) <)@ 		!sc8885 5 5 5 5M 5 5 5p MCk222	 bgag$$+ + +( ( (            l l l l l} l l l^ xV3' 3' 3' 3' 3' 3' 3' 3'l 		Cg&&&% % % % % % % %P 
rufQh"%'	9	9	9*' *' *' *' *'- *' *' *'Z +s
3
3
3
 
 
 
 
: 
 
 
	 	 	 	 	X 	 	 	    |" |" |" |" |"} |" |" |"~ x#6***@& @& @& @& @&M @& @& @&F MCk222	4# 4# 4# 4# 4#= 4# 4# 4#n <#:666x" x" x" x" x"} x" x" x"v x#F###M- M- M- M- M- M- M- M-` 
c	)	)	)H H H H Hx H H HV x#F###0" 0" 0" 0" 0" 0" 0" 0"f 
	"	"	"A A A A Am A A AH g%   @ @ @ @ @} @ @ @F x#F###2# 2# 2# 2# 2# 2# 2# 2#j 
rufH	5	5	566 66 66 66 66 66 66 66r 
	"	"	"8  8  8  8  8 = 8  8  8 v <Z(((j( j( j( j( j( j( j( j(Z 		Cg&&&K! K! K! K! K!M K! K! K!\ M{+++	?; ?; ?; ?; ?;M ?; ?; ?;D MCk222	23 23 23 23 23= 23 23 23j <#J///J J J J Jm J J JZ o-888&. &. &. &. &.] &. &. &.R ^c555
X X X X XM X X Xv 
ECc7 7 7 7 7= 7 7 7t <#J///PF PF PF PF PFm PF PF PFf o-888h! h! h! h! h!= h! h! h!V ('-?@@@ D4 D4 D4 D4 D4m D4 D4 D4N o-8885 5 5 5 5m 5 5 5p o=111u u u u uM u u up MCk222	28 28 28 28 28= 28 28 28j <#J///O" O" O" O" O"] O" O" O"d ^...
& & &^` ` ` ` ` ` ` `F 		Cg&&&30 30 30 30 30 30 30 30l 
c	)	)	)QF QF QF QF QF= QF QF QFh <#J////% /% /% /% /%- /% /% /%d &%2CDDDL L L L L L L L^ "!777'9 '9 '9 '9 '9= '9 '9 '9T <#J///* * *| | | | |= | | |~ <Z(((I I I I I= I I IX <Z(((( ( ( ( (] ( ( (V ^c555
2 2 2 2 2} 2 2 2j  #N;;;0) 0) 0) 0) 0)= 0) 0) 0)f <#J///% % % % %M % % %P M{+++	6 6 6 6 6] 6 6 6r ^cS|<<<
G7 G7 G7 G7 G7= G7 G7 G7T <#J///s& s& s& s& s&= s& s& s&l <#J///t t t t tm t t tn	 o-888M2 M2 M2 M2 M2} M2 M2 M2`  ^444@F @F @F @F @F] @F @F @FF ^al333
/A /A /A /A /AM /A /A /Ad MC3[999	// // // // //M // // //d M{+++	E E E E E- E E EP +9
%
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