
    geV                        d dl mZ d dlmZ d dlmZmZmZmZ	m
Z
 d dlmZmZ d dlmZ d dlmZmZmZmZmZmZmZmZmZmZ d dlZdd	lmZmZmZm Z  d dl!m"c m#Z# dd
l$m%Z%m&Z&m'Z' d Z( G d de          Z) e)d          Z* G d de)          Z+ e+dd          Z, G d de          Z- e-d          Z. G d de          Z/ e/d          Z0 G d de          Z1 e1ddd          Z2 G d d e          Z3 e3d!          Z4 G d" d#e          Z5 e5d$          Z6 G d% d&e          Z7 e7dd'd(          Z8 G d) d*e          Z9 e9d+d,-          Z: G d. d/e          Z; e;d d0d1          Z< G d2 d3e          Z= e=d4d d56          Z> G d7 d8e          Z? e?d9d:-          Z@ G d; d<e          ZA eAdd=d>          ZBd? ZCd@ ZDdA ZE G dB dCe          ZF eFddDdE          ZG G dF dGe          ZH eHejI         dHdI          ZJ G dJ dKe          ZK eKejI         dLdM          ZL G dN dOe          ZM eMdPdQ          ZNdR ZO G dS dTe          ZP G dU dVeP          ZQ eQdWdX-          ZR G dY dZeP          ZS eSd[d\-          ZT eU eV            W                                X                                          ZY eeYe          \  ZZZ[eZe[z   Z\dS )]    )partial)special)entr	logsumexpbetalngammalnzeta)
_lazywhererng_integers)interp1d)
floorceillogexpsqrtlog1pexpm1tanhcoshsinhN   )rv_discreteget_distribution_names_check_shape
_ShapeInfo)_PyFishersNCHypergeometric_PyWalleniusNCHypergeometric_PyStochasticLib3c                 2    | t          j        |           k    S N)nproundxs    <lib/python3.11/site-packages/scipy/stats/_discrete_distns.py_isintegralr&      s        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dZd ZdS )	binom_gena  A binomial discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `binom` is:

    .. math::

       f(k) = \binom{n}{k} p^k (1-p)^{n-k}

    for :math:`k \in \{0, 1, \dots, n\}`, :math:`0 \leq p \leq 1`

    `binom` takes :math:`n` and :math:`p` as shape parameters,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.

    %(after_notes)s

    %(example)s

    See Also
    --------
    hypergeom, nbinom, nhypergeom

    c                 b    t          dddt          j        fd          t          dddd          gS 	NnTr   TFpFr   r   TTr   r!   infselfs    r%   _shape_infozbinom_gen._shape_info8   4    3q"&k=AA3v|<<> 	>r'   Nc                 0    |                     |||          S r    )binomialr4   r,   r.   sizerandom_states        r%   _rvszbinom_gen._rvs<   s    $$Q4000r'   c                 J    |dk    t          |          z  |dk    z  |dk    z  S Nr   r   r&   r4   r,   r.   s      r%   	_argcheckzbinom_gen._argcheck?   s)    Q+a..(AF3qAv>>r'   c                     | j         |fS r    ar@   s      r%   _get_supportzbinom_gen._get_supportB   s    vqyr'   c                     t          |          }t          |dz             t          |dz             t          ||z
  dz             z   z
  }|t          j        ||          z   t          j        ||z
  |           z   S Nr   )r   gamlnr   xlogyxlog1py)r4   r$   r,   r.   kcombilns         r%   _logpmfzbinom_gen._logpmfE   sl    !HH1::qseAaCEll!:;q!,,,wqsQB/G/GGGr'   c                 .    t          j        |||          S r    )_boost
_binom_pdfr4   r$   r,   r.   s       r%   _pmfzbinom_gen._pmfJ   s     Aq)))r'   c                 L    t          |          }t          j        |||          S r    )r   rO   
_binom_cdfr4   r$   r,   r.   rK   s        r%   _cdfzbinom_gen._cdfN   "    !HH Aq)))r'   c                 L    t          |          }t          j        |||          S r    )r   rO   	_binom_sfrU   s        r%   _sfzbinom_gen._sfR   s"    !HH1a(((r'   c                 .    t          j        |||          S r    )rO   
_binom_isfrQ   s       r%   _isfzbinom_gen._isfV        Aq)))r'   c                 .    t          j        |||          S r    )rO   
_binom_ppfr4   qr,   r.   s       r%   _ppfzbinom_gen._ppfY   r^   r'   mvc                     t          j        ||          }t          j        ||          }d\  }}d|v rt          j        ||          }d|v rt          j        ||          }||||fS )NNNsrK   )rO   _binom_mean_binom_variance_binom_skewness_binom_kurtosis_excess)r4   r,   r.   momentsmuvarg1g2s           r%   _statszbinom_gen._stats\   ss    1%%$Q**B'>>'1--B'>>.q!44B3Br'   c                     t           j        d|dz            }|                     |||          }t          j        t	          |          d          S )Nr   r   axis)r!   r_rR   sumr   )r4   r,   r.   rK   valss        r%   _entropyzbinom_gen._entropyf   sE    E!AE'NyyAq!!vd4jjq))))r'   rf   rd   __name__
__module____qualname____doc__r5   r<   rA   rE   rM   rR   rV   rZ   r]   rc   rq   rx    r'   r%   r)   r)      s         6> > >1 1 1 1? ? ?  H H H
* * ** * *) ) )* * ** * *   * * * * *r'   r)   binom)namec                   \    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dS )bernoulli_gena  A Bernoulli discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `bernoulli` is:

    .. math::

       f(k) = \begin{cases}1-p  &\text{if } k = 0\\
                           p    &\text{if } k = 1\end{cases}

    for :math:`k` in :math:`\{0, 1\}`, :math:`0 \leq p \leq 1`

    `bernoulli` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.

    %(after_notes)s

    %(example)s

    c                 (    t          dddd          gS Nr.   Fr/   r0   r   r3   s    r%   r5   zbernoulli_gen._shape_info       3v|<<==r'   Nc                 @    t                               | d|||          S )Nr   r:   r;   )r)   r<   r4   r.   r:   r;   s       r%   r<   zbernoulli_gen._rvs   s    ~~dAqt,~OOOr'   c                     |dk    |dk    z  S r>   r   r4   r.   s     r%   rA   zbernoulli_gen._argcheck   s    Q16""r'   c                     | j         | j        fS r    )rD   br   s     r%   rE   zbernoulli_gen._get_support   s    vtv~r'   c                 :    t                               |d|          S rG   )r   rM   r4   r$   r.   s      r%   rM   zbernoulli_gen._logpmf   s    }}Q1%%%r'   c                 :    t                               |d|          S rG   )r   rR   r   s      r%   rR   zbernoulli_gen._pmf   s     zz!Q"""r'   c                 :    t                               |d|          S rG   )r   rV   r   s      r%   rV   zbernoulli_gen._cdf       zz!Q"""r'   c                 :    t                               |d|          S rG   )r   rZ   r   s      r%   rZ   zbernoulli_gen._sf   s    yyAq!!!r'   c                 :    t                               |d|          S rG   )r   r]   r   s      r%   r]   zbernoulli_gen._isf   r   r'   c                 :    t                               |d|          S rG   )r   rc   )r4   rb   r.   s      r%   rc   zbernoulli_gen._ppf   r   r'   c                 8    t                               d|          S rG   )r   rq   r   s     r%   rq   zbernoulli_gen._stats   s    ||Aq!!!r'   c                 F    t          |          t          d|z
            z   S rG   )r   r   s     r%   rx   zbernoulli_gen._entropy   s    Awwac""r'   rf   rz   r   r'   r%   r   r   o   s         0> > >P P P P# # #  & & &# # #
# # #" " "# # ## # #" " "# # # # #r'   r   	bernoulli)r   r   c                   @    e Zd ZdZd ZddZd Zd Zd Zd Z	dd
Z
dS )betabinom_gena  A beta-binomial discrete random variable.

    %(before_notes)s

    Notes
    -----
    The beta-binomial distribution is a binomial distribution with a
    probability of success `p` that follows a beta distribution.

    The probability mass function for `betabinom` is:

    .. math::

       f(k) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}

    for :math:`k \in \{0, 1, \dots, n\}`, :math:`n \geq 0`, :math:`a > 0`,
    :math:`b > 0`, where :math:`B(a, b)` is the beta function.

    `betabinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution

    %(after_notes)s

    .. versionadded:: 1.4.0

    See Also
    --------
    beta, binom

    %(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,   Tr   r-   rD   FFFr   r1   r3   s    r%   r5   zbetabinom_gen._shape_info   sS    3q"&k=AA326{NCC326{NCCE 	Er'   Nc                 ^    |                     |||          }|                    |||          S r    )betar8   )r4   r,   rD   r   r:   r;   r.   s          r%   r<   zbetabinom_gen._rvs   s1    aD))$$Q4000r'   c                 
    d|fS Nr   r   r4   r,   rD   r   s       r%   rE   zbetabinom_gen._get_support       !tr'   c                 J    |dk    t          |          z  |dk    z  |dk    z  S r   r?   r   s       r%   rA   zbetabinom_gen._argcheck   s)    Q+a..(AE2a!e<<r'   c                     t          |          }t          |dz              t          ||z
  dz   |dz             z
  }|t          ||z   ||z
  |z             z   t          ||          z
  S rG   )r   r   r   )r4   r$   r,   rD   r   rK   rL   s          r%   rM   zbetabinom_gen._logpmf   sg    !HHq1u::+q1uqy!a% 8 88Aq1uqy111F1aLL@@r'   c                 L    t          |                     ||||                    S r    r   rM   )r4   r$   r,   rD   r   s        r%   rR   zbetabinom_gen._pmf   s"    4<<1a++,,,r'   rd   c                 J   |||z   z  }d|z
  }||z  }|||z   |z   z  |z  |z  ||z   dz   z  }d\  }	}
d|v r7dt          |          z  }	|	||z   d|z  z   ||z
  z  z  }	|	||z   dz   ||z   z  z  }	d|v r||z                       |j                  }
|
||z   dz
  d|z  z   z  }
|
d|z  |z  |dz
  z  z  }
|
d|dz  z  z  }
|
d|z  |z  |z  d|z
  z  z  }
|
d	|z  |z  |dz  z  z  }
|
||z   dz  d|z   |z   z  z  }
|
||z  |z  ||z   dz   z  ||z   dz   z  ||z   |z   z  z  }
|
dz  }
|||	|
fS )
Nr   rf   rg         ?   rK            )r   astypedtype)r4   r,   rD   r   rl   e_pe_qrm   rn   ro   rp   s              r%   rq   zbetabinom_gen._stats   s   1q5k#gW1q519o#c)QUQY7B'>>tCyyB1q51q5=QU++B1q519Q''B'>>a%	**B1q519q1u$%B!a%!)q1u%%B!a1f*B!c'A+/QU++B"s(S.16))B1q5Q,!a%!),,B1q519A	*a!eai8AEAIFGB!GB3Br'   rf   ry   )r{   r|   r}   r~   r5   r<   rE   rA   rM   rR   rq   r   r'   r%   r   r      s        " "FE E E
1 1 1 1  = = =A A A
- - -     r'   r   	betabinomc                   V    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S )
nbinom_gena  A negative binomial discrete random variable.

    %(before_notes)s

    Notes
    -----
    Negative binomial distribution describes a sequence of i.i.d. Bernoulli
    trials, repeated until a predefined, non-random number of successes occurs.

    The probability mass function of the number of failures for `nbinom` is:

    .. math::

       f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k

    for :math:`k \ge 0`, :math:`0 < p \leq 1`

    `nbinom` takes :math:`n` and :math:`p` as shape parameters where :math:`n`
    is the number of successes, :math:`p` is the probability of a single
    success, and :math:`1-p` is the probability of a single failure.

    Another common parameterization of the negative binomial distribution is
    in terms of the mean number of failures :math:`\mu` to achieve :math:`n`
    successes. The mean :math:`\mu` is related to the probability of success
    as

    .. math::

       p = \frac{n}{n + \mu}

    The number of successes :math:`n` may also be specified in terms of a
    "dispersion", "heterogeneity", or "aggregation" parameter :math:`\alpha`,
    which relates the mean :math:`\mu` to the variance :math:`\sigma^2`,
    e.g. :math:`\sigma^2 = \mu + \alpha \mu^2`. Regardless of the convention
    used for :math:`\alpha`,

    .. math::

       p &= \frac{\mu}{\sigma^2} \\
       n &= \frac{\mu^2}{\sigma^2 - \mu}

    %(after_notes)s

    %(example)s

    See Also
    --------
    hypergeom, binom, nhypergeom

    c                 b    t          dddt          j        fd          t          dddd          gS r+   r1   r3   s    r%   r5   znbinom_gen._shape_info;  r6   r'   Nc                 0    |                     |||          S r    )negative_binomialr9   s        r%   r<   znbinom_gen._rvs?  s    --aD999r'   c                 *    |dk    |dk    z  |dk    z  S r>   r   r@   s      r%   rA   znbinom_gen._argcheckB  s    A!a% AF++r'   c                 .    t          j        |||          S r    )rO   _nbinom_pdfrQ   s       r%   rR   znbinom_gen._pmfE  s    !!Q***r'   c                     t          ||z             t          |dz             z
  t          |          z
  }||t          |          z  z   t          j        ||           z   S rG   )rH   r   r   rJ   )r4   r$   r,   r.   coeffs        r%   rM   znbinom_gen._logpmfI  sS    ac

U1Q3ZZ'%((2qQx'/!aR"8"888r'   c                 L    t          |          }t          j        |||          S r    )r   rO   _nbinom_cdfrU   s        r%   rV   znbinom_gen._cdfM  s"    !HH!!Q***r'   c                 D   t          |          }|                     |||          }|dk    }d }|}t          j        d          5   |||         ||         ||                   ||<   t          j        ||                    || <   d d d            n# 1 swxY w Y   |S )N      ?c                 `    t          j        t          j        | dz   |d|z
                       S rG   )r!   r   r   betainc)rK   r,   r.   s      r%   f1znbinom_gen._logcdf.<locals>.f1V  s+    8W_QUAq1u===>>>r'   ignore)divide)r   rV   r!   errstater   )	r4   r$   r,   r.   rK   cdfcondr   logcdfs	            r%   _logcdfznbinom_gen._logcdfQ  s    !HHii1a  Sy	? 	? 	? [))) 	/ 	/2agqw$88F4LF3u:..FD5M	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ s   ABBBc                 L    t          |          }t          j        |||          S r    )r   rO   
_nbinom_sfrU   s        r%   rZ   znbinom_gen._sf`  rW   r'   c                     t          j        d          5  t          j        |||          cd d d            S # 1 swxY w Y   d S Nr   over)r!   r   rO   _nbinom_isfrQ   s       r%   r]   znbinom_gen._isfd      [h''' 	/ 	/%aA..	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/   9= =c                     t          j        d          5  t          j        |||          cd d d            S # 1 swxY w Y   d S r   )r!   r   rO   _nbinom_ppfra   s       r%   rc   znbinom_gen._ppfh  r   r   c                     t          j        ||          t          j        ||          t          j        ||          t          j        ||          fS r    )rO   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessr@   s      r%   rq   znbinom_gen._statsl  sL    1%%#Aq))#Aq))*1a00	
 	
r'   rf   )r{   r|   r}   r~   r5   r<   rA   rR   rM   rV   r   rZ   r]   rc   rq   r   r'   r%   r   r     s        1 1d> > >: : : :, , ,+ + +9 9 9+ + +  * * */ / // / /
 
 
 
 
r'   r   nbinomc                   V    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S )geom_gena  A geometric discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `geom` is:

    .. math::

        f(k) = (1-p)^{k-1} p

    for :math:`k \ge 1`, :math:`0 < p \leq 1`

    `geom` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.

    %(after_notes)s

    See Also
    --------
    planck

    %(example)s

    c                 (    t          dddd          gS r   r   r3   s    r%   r5   zgeom_gen._shape_info  r   r'   Nc                 0    |                     ||          S Nr:   )	geometricr   s       r%   r<   zgeom_gen._rvs  s    %%ad%333r'   c                     |dk    |dk    z  S Nr   r   r   r   s     r%   rA   zgeom_gen._argcheck  s    Q1q5!!r'   c                 >    t          j        d|z
  |dz
            |z  S rG   )r!   powerr4   rK   r.   s      r%   rR   zgeom_gen._pmf  s!    x!QqS!!A%%r'   c                 T    t          j        |dz
  |           t          |          z   S rG   )r   rJ   r   r   s      r%   rM   zgeom_gen._logpmf  s%    q1uqb))CFF22r'   c                 b    t          |          }t          t          |           |z             S r    )r   r   r   r4   r$   r.   rK   s       r%   rV   zgeom_gen._cdf  s*    !HHeQBiik""""r'   c                 R    t          j        |                     ||                    S r    )r!   r   _logsfr   s      r%   rZ   zgeom_gen._sf  s     vdkk!Q''(((r'   c                 F    t          |          }|t          |           z  S r    )r   r   r   s       r%   r   zgeom_gen._logsf  s    !HHr{r'   c                     t          t          |           t          |           z            }|                     |dz
  |          }t          j        ||k    |dk    z  |dz
  |          S r   )r   r   rV   r!   where)r4   rb   r.   rw   temps        r%   rc   zgeom_gen._ppf  s`    E1"IIqb		)**yya##xtax0$q&$???r'   c                     d|z  }d|z
  }||z  |z  }d|z
  t          |          z  }t          j        g d|          d|z
  z  }||||fS )Nr          @)r   ir   )r   r!   polyval)r4   r.   rm   qrrn   ro   rp   s          r%   rq   zgeom_gen._stats  sa    UU1fqj!etBxxZ


A&&A.3Br'   c                 j    t          j        |           t          j        |           d|z
  z  |z  z
  S Nr   )r!   r   r   r   s     r%   rx   zgeom_gen._entropy  s/    q		zBHaRLLCE2Q666r'   rf   )r{   r|   r}   r~   r5   r<   rA   rR   rM   rV   rZ   r   rc   rq   rx   r   r'   r%   r   r   x  s         8> > >4 4 4 4" " "& & &3 3 3# # #) ) )  @ @ @
  7 7 7 7 7r'   r   geomzA geometric)rD   r   longnamec                   \    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dS )hypergeom_gena  A hypergeometric discrete random variable.

    The hypergeometric distribution models drawing objects from a bin.
    `M` is the total number of objects, `n` is total number of Type I objects.
    The random variate represents the number of Type I objects in `N` drawn
    without replacement from the total population.

    %(before_notes)s

    Notes
    -----
    The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not
    universally accepted.  See the Examples for a clarification of the
    definitions used here.

    The probability mass function is defined as,

    .. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}
                                   {\binom{M}{N}}

    for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial
    coefficients are defined as,

    .. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import hypergeom
    >>> import matplotlib.pyplot as plt

    Suppose we have a collection of 20 animals, of which 7 are dogs.  Then if
    we want to know the probability of finding a given number of dogs if we
    choose at random 12 of the 20 animals, we can initialize a frozen
    distribution and plot the probability mass function:

    >>> [M, n, N] = [20, 7, 12]
    >>> rv = hypergeom(M, n, N)
    >>> x = np.arange(0, n+1)
    >>> pmf_dogs = rv.pmf(x)

    >>> fig = plt.figure()
    >>> ax = fig.add_subplot(111)
    >>> ax.plot(x, pmf_dogs, 'bo')
    >>> ax.vlines(x, 0, pmf_dogs, lw=2)
    >>> ax.set_xlabel('# of dogs in our group of chosen animals')
    >>> ax.set_ylabel('hypergeom PMF')
    >>> plt.show()

    Instead of using a frozen distribution we can also use `hypergeom`
    methods directly.  To for example obtain the cumulative distribution
    function, use:

    >>> prb = hypergeom.cdf(x, M, n, N)

    And to generate random numbers:

    >>> R = hypergeom.rvs(M, n, N, size=10)

    See Also
    --------
    nhypergeom, binom, nbinom

    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 )NMTr   r-   r,   Nr1   r3   s    r%   r5   zhypergeom_gen._shape_info  S    3q"&k=AA3q"&k=AA3q"&k=AAC 	Cr'   Nc                 :    |                     |||z
  ||          S r   )hypergeometric)r4   r   r,   r   r:   r;   s         r%   r<   zhypergeom_gen._rvs  s#    **1ac14*@@@r'   c                 b    t          j        |||z
  z
  d          t          j        ||          fS r   r!   maximumminimum)r4   r   r,   r   s       r%   rE   zhypergeom_gen._get_support  s-    z!QqS'1%%rz!Q'7'777r'   c                     |dk    |dk    z  |dk    z  }|||k    ||k    z  z  }|t          |          t          |          z  t          |          z  z  }|S r   r?   )r4   r   r,   r   r   s        r%   rA   zhypergeom_gen._argcheck  s_    A!q&!Q!V,aAF##AQ/+a..@@r'   c                 6   ||}}||z
  }t          |dz   d          t          |dz   d          z   t          ||z
  dz   |dz             z   t          |dz   ||z
  dz             z
  t          ||z
  dz   ||z
  |z   dz             z
  t          |dz   d          z
  }|S rG   r   )	r4   rK   r   r,   r   totgoodbadresults	            r%   rM   zhypergeom_gen._logpmf  s    qTDja##fSUA&6&66Aa19M9MM1d1fQh''(*01QAa	*B*BCQ""# r'   c                 0    t          j        ||||          S r    )rO   _hypergeom_pdfr4   rK   r   r,   r   s        r%   rR   zhypergeom_gen._pmf      $Q1a000r'   c                 0    t          j        ||||          S r    )rO   _hypergeom_cdfr  s        r%   rV   zhypergeom_gen._cdf"  r  r'   c                 x   d|z  d|z  d|z  }}}||z
  }||dz   z  d|z  ||z
  z  z
  d|z  |z  z
  }||dz
  |z  |z  z  }|d|z  |z  ||z
  z  |z  d|z  dz
  z  z  }|||z  ||z
  z  |z  |dz
  z  |dz
  z  z  }t          j        |||          t          j        |||          t          j        |||          |fS )Nr   r   g      @g      @r   r         @)rO   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r4   r   r,   r   mrp   s         r%   rq   zhypergeom_gen._stats%  s   q&"q&"q&a1E !a%[26QU++b1fqj8
q1ukAo
b1fqjAE"Q&"q&1*55
a!eq1uo!QV,B77"1a++&q!Q//&q!Q//	
 	
r'   c                     t           j        |||z
  z
  t          ||          dz            }|                     ||||          }t          j        t          |          d          S )Nr   r   rs   )r!   ru   minpmfrv   r   )r4   r   r,   r   rK   rw   s         r%   rx   zhypergeom_gen._entropy6  sY    E!q1u+c!Qii!m+,xx1a##vd4jjq))))r'   c                 0    t          j        ||||          S r    )rO   _hypergeom_sfr  s        r%   rZ   zhypergeom_gen._sf;  s    #Aq!Q///r'   c                    g }t          t          j        ||||           D ]\  }}}}	|dz   |dz   z  |dz
  |	dz
  z  k     rG|                    t	          t          |                     ||||	                                          ft          j        |dz   |	dz             }
|                    t          | 	                    |
|||	                               t          j
        |          S )Nr   r   )zipr!   broadcast_arraysappendr   r   r   aranger   rM   asarrayr4   rK   r   r,   r   resquantr  r	  drawk2s              r%   r   zhypergeom_gen._logsf>  s    &)2+>q!Q+J+J&K 	I 	I"E3dc	*dSjTCZ-HHH

5#dkk%dD&I&I"J"J!JKKLLLL Yuqy$(33

9T\\"c4%F%FGGHHHHz#r'   c                    g }t          t          j        ||||           D ]\  }}}}	|dz   |dz   z  |dz
  |	dz
  z  k    rG|                    t	          t          |                     ||||	                                          ft          j        d|dz             }
|                    t          | 	                    |
|||	                               t          j
        |          S )Nr   r   r   )r  r!   r  r   r   r   logsfr!  r   rM   r"  r#  s              r%   r   zhypergeom_gen._logcdfJ  s    &)2+>q!Q+J+J&K 	I 	I"E3dc	*dSjTCZ-HHH

5#djjT4&H&H"I"I!IJJKKKK Yq%!),,

9T\\"c4%F%FGGHHHHz#r'   rf   )r{   r|   r}   r~   r5   r<   rE   rA   rM   rR   rV   rq   rx   rZ   r   r   r   r'   r%   r   r     s        A ADC C C
A A A A8 8 8    1 1 11 1 1
 
 
"* * *
0 0 0
 
 

 
 
 
 
r'   r   	hypergeomc                   >    e Zd ZdZd Zd Zd Zd
dZd Zd Z	d	 Z
dS )nhypergeom_genab  A negative hypergeometric discrete random variable.

    Consider a box containing :math:`M` balls:, :math:`n` red and
    :math:`M-n` blue. We randomly sample balls from the box, one
    at a time and *without* replacement, until we have picked :math:`r`
    blue balls. `nhypergeom` is the distribution of the number of
    red balls :math:`k` we have picked.

    %(before_notes)s

    Notes
    -----
    The symbols used to denote the shape parameters (`M`, `n`, and `r`) are not
    universally accepted. See the Examples for a clarification of the
    definitions used here.

    The probability mass function is defined as,

    .. math:: f(k; M, n, r) = \frac{{{k+r-1}\choose{k}}{{M-r-k}\choose{n-k}}}
                                   {{M \choose n}}

    for :math:`k \in [0, n]`, :math:`n \in [0, M]`, :math:`r \in [0, M-n]`,
    and the binomial coefficient is:

    .. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.

    It is equivalent to observing :math:`k` successes in :math:`k+r-1`
    samples with :math:`k+r`'th sample being a failure. The former
    can be modelled as a hypergeometric distribution. The probability
    of the latter is simply the number of failures remaining
    :math:`M-n-(r-1)` divided by the size of the remaining population
    :math:`M-(k+r-1)`. This relationship can be shown as:

    .. math:: NHG(k;M,n,r) = HG(k;M,n,k+r-1)\frac{(M-n-(r-1))}{(M-(k+r-1))}

    where :math:`NHG` is probability mass function (PMF) of the
    negative hypergeometric distribution and :math:`HG` is the
    PMF of the hypergeometric distribution.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import nhypergeom
    >>> import matplotlib.pyplot as plt

    Suppose we have a collection of 20 animals, of which 7 are dogs.
    Then if we want to know the probability of finding a given number
    of dogs (successes) in a sample with exactly 12 animals that
    aren't dogs (failures), we can initialize a frozen distribution
    and plot the probability mass function:

    >>> M, n, r = [20, 7, 12]
    >>> rv = nhypergeom(M, n, r)
    >>> x = np.arange(0, n+2)
    >>> pmf_dogs = rv.pmf(x)

    >>> fig = plt.figure()
    >>> ax = fig.add_subplot(111)
    >>> ax.plot(x, pmf_dogs, 'bo')
    >>> ax.vlines(x, 0, pmf_dogs, lw=2)
    >>> ax.set_xlabel('# of dogs in our group with given 12 failures')
    >>> ax.set_ylabel('nhypergeom PMF')
    >>> plt.show()

    Instead of using a frozen distribution we can also use `nhypergeom`
    methods directly.  To for example obtain the probability mass
    function, use:

    >>> prb = nhypergeom.pmf(x, M, n, r)

    And to generate random numbers:

    >>> R = nhypergeom.rvs(M, n, r, size=10)

    To verify the relationship between `hypergeom` and `nhypergeom`, use:

    >>> from scipy.stats import hypergeom, nhypergeom
    >>> M, n, r = 45, 13, 8
    >>> k = 6
    >>> nhypergeom.pmf(k, M, n, r)
    0.06180776620271643
    >>> hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))
    0.06180776620271644

    See Also
    --------
    hypergeom, binom, nbinom

    References
    ----------
    .. [1] Negative Hypergeometric Distribution on Wikipedia
           https://en.wikipedia.org/wiki/Negative_hypergeometric_distribution

    .. [2] Negative Hypergeometric Distribution from
           http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Negativehypergeometric.pdf

    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   Tr   r-   r,   rr1   r3   s    r%   r5   znhypergeom_gen._shape_info  r   r'   c                 
    d|fS r   r   )r4   r   r,   r.  s       r%   rE   znhypergeom_gen._get_support  r   r'   c                     |dk    ||k    z  |dk    z  |||z
  k    z  }|t          |          t          |          z  t          |          z  z  }|S r   r?   )r4   r   r,   r.  r   s        r%   rA   znhypergeom_gen._argcheck  sU    Q16"a1f-ac:AQ/+a..@@r'   Nc                 H     t            fd            } ||||||          S )Nc                 \                        | ||          \  }}t          j        ||dz             }                    || ||          }t	          ||dd          }	 |	|                    |                                        t                    }
||
                                S |
S )Nr   nextextrapolate)kind
fill_valuer   )	supportr!   r!  r   r   uniformr   intitem)r   r,   r.  r:   r;   rD   r   ksr   ppfrvsr4   s              r%   _rvs1z"nhypergeom_gen._rvs.<locals>._rvs1  s     <<1a((DAq1ac""B((2q!Q''C3MJJJC#l***5566==cBBC|xxzz!Jr'   r   _vectorize_rvs_over_shapes)r4   r   r,   r.  r:   r;   r>  s   `      r%   r<   znhypergeom_gen._rvs  sD    	#		 		 		 		 
$	#		 uQ14lCCCCr'   c                 R    |dk    |dk    z  }t          | ||||fd d          }|S )Nr   c                 "   t          | dz   |           t          | |z   d          z   t          || z
  dz   ||z
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  t          ||z
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  dz   d          z   t          |dz   ||z
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  S rG   r  )rK   r   r,   r.  s       r%   <lambda>z(nhypergeom_gen._logpmf.<locals>.<lambda>  s    "(1a..6!A#q>>!A!'!Aqs1uQw!7!7"8:@1Qq!:L:L"M!'!QqSU!3!3"46<QqS!nn"E r'           )	fillvalue)r
   )r4   rK   r   r,   r.  r   r  s          r%   rM   znhypergeom_gen._logpmf  sN    aAF#TEAq!Q<F F '*+ + + r'   c                 L    t          |                     ||||                    S r    r   )r4   rK   r   r,   r.  s        r%   rR   znhypergeom_gen._pmf  s$     4<<1a++,,,r'   c                     d|z  d|z  d|z  }}}||z  ||z
  dz   z  }||dz   z  |z  ||z
  dz   ||z
  dz   z  z  d|||z
  dz   z  z
  z  }d\  }}||||fS )Nr   r   r   rf   r   )r4   r   r,   r.  rm   rn   ro   rp   s           r%   rq   znhypergeom_gen._stats  s     Q$1bda1qSAaCE]1gaiAaCEAaCE?+q1!A;? B3Br'   rf   )r{   r|   r}   r~   r5   rE   rA   r<   rM   rR   rq   r   r'   r%   r,  r,  Z  s        b bHC C C
    
D D D D   - - -
    r'   r,  
nhypergeomc                   2    e Zd ZdZd ZddZd Zd Zd ZdS )	
logser_gena  A Logarithmic (Log-Series, Series) discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `logser` is:

    .. math::

        f(k) = - \frac{p^k}{k \log(1-p)}

    for :math:`k \ge 1`, :math:`0 < p < 1`

    `logser` takes :math:`p` as shape parameter,
    where :math:`p` is the probability of a single success
    and :math:`1-p` is the probability of a single failure.

    %(after_notes)s

    %(example)s

    c                 (    t          dddd          gS r   r   r3   s    r%   r5   zlogser_gen._shape_info  r   r'   Nc                 0    |                     ||          S r   )	logseriesr   s       r%   r<   zlogser_gen._rvs  s     %%ad%333r'   c                     |dk    |dk     z  S r>   r   r   s     r%   rA   zlogser_gen._argcheck  s    A!a%  r'   c                 f    t          j        ||           dz  |z  t          j        |           z  S r   )r!   r   r   r   r   s      r%   rR   zlogser_gen._pmf!  s/    A$q(7=!+<+<<<r'   c                    t          j        |           }||dz
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  dz  z  }|d|z  |z  z
  d|dz  z  z   }|t          j        |d          z  }| |z  d|dz
  dz  z  d|z  |dz
  dz  z  z
  d|z  |z  |dz
  dz  z  z   z  }	|	d|z  |z  z
  d|z  |z  |z  z   d|dz  z  z
  }
|
|dz  z  dz
  }||||fS )	Nr   r   r         ?r   r      r  )r   r   r!   r   )r4   r.   r.  rm   mu2prn   mu3pmu3ro   mu4pmu4rp   s               r%   rq   zlogser_gen._stats%  s@   M1"!c']QrAvS1$RUlrAvQ37Q,.QrT$Y2q5(28C%%%rAv1Q3(NQqSAEA:--!A1q0@@BQtVBY42-"a%736\C3Br'   rf   )	r{   r|   r}   r~   r5   r<   rA   rR   rq   r   r'   r%   rJ  rJ    sn         0> > >4 4 4 4
! ! != = =    r'   rJ  logserzA logarithmicc                   J    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S )poisson_gena  A Poisson discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `poisson` is:

    .. math::

        f(k) = \exp(-\mu) \frac{\mu^k}{k!}

    for :math:`k \ge 0`.

    `poisson` takes :math:`\mu \geq 0` as shape parameter.
    When :math:`\mu = 0`, the ``pmf`` method
    returns ``1.0`` at quantile :math:`k = 0`.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS )Nrm   Fr   r-   r1   r3   s    r%   r5   zpoisson_gen._shape_infoQ  s    4BF]CCDDr'   c                     |dk    S r   r   )r4   rm   s     r%   rA   zpoisson_gen._argcheckU  s    Qwr'   Nc                 .    |                     ||          S r    poisson)r4   rm   r:   r;   s       r%   r<   zpoisson_gen._rvsX  s    ##B---r'   c                 \    t          j        ||          t          |dz             z
  |z
  }|S rG   )r   rI   rH   )r4   rK   rm   Pks       r%   rM   zpoisson_gen._logpmf[  s,    ]1b!!E!a%LL025	r'   c                 H    t          |                     ||                    S r    r   )r4   rK   rm   s      r%   rR   zpoisson_gen._pmf_  s    4<<2&&'''r'   c                 J    t          |          }t          j        ||          S r    )r   r   pdtrr4   r$   rm   rK   s       r%   rV   zpoisson_gen._cdfc  s    !HH|Ar"""r'   c                 J    t          |          }t          j        ||          S r    )r   r   pdtrcre  s       r%   rZ   zpoisson_gen._sfg  s    !HH}Q###r'   c                     t          t          j        ||                    }t          j        |dz
  d          }t          j        ||          }t          j        ||k    ||          S r   )r   r   pdtrikr!   r  rd  r   )r4   rb   rm   rw   vals1r   s         r%   rc   zpoisson_gen._ppfk  sY    GN1b))**
4!8Q''|E2&&x	5$///r'   c                     |}t          j        |          }|dk    }t          ||fd t           j                  }t          ||fd t           j                  }||||fS )Nr   c                 &    t          d| z            S r   r   r#   s    r%   rC  z$poisson_gen._stats.<locals>.<lambda>u  s    d3q5kk r'   c                     d| z  S r   r   r#   s    r%   rC  z$poisson_gen._stats.<locals>.<lambda>v  s
    c!e r'   )r!   r"  r
   r2   )r4   rm   rn   tmp
mu_nonzeroro   rp   s          r%   rq   zpoisson_gen._statsq  s_    jnn1W

SF,A,A26JJ
SFOORVDD3Br'   rf   )r{   r|   r}   r~   r5   rA   r<   rM   rR   rV   rZ   rc   rq   r   r'   r%   rZ  rZ  8  s         0E E E  . . . .  ( ( (# # #$ $ $0 0 0    r'   rZ  r_  z	A Poisson)r   r   c                   P    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d Zd	S )
planck_gena  A Planck discrete exponential random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `planck` is:

    .. math::

        f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)

    for :math:`k \ge 0` and :math:`\lambda > 0`.

    `planck` takes :math:`\lambda` as shape parameter. The Planck distribution
    can be written as a geometric distribution (`geom`) with
    :math:`p = 1 - \exp(-\lambda)` shifted by ``loc = -1``.

    %(after_notes)s

    See Also
    --------
    geom

    %(example)s

    c                 @    t          dddt          j        fd          gS )NlambdaFr   r   r1   r3   s    r%   r5   zplanck_gen._shape_info  s    8UQKHHIIr'   c                     |dk    S r   r   )r4   lambda_s     r%   rA   zplanck_gen._argcheck  s    {r'   c                 L    t          |            t          | |z            z  S r    )r   r   )r4   rK   rv  s      r%   rR   zplanck_gen._pmf  s$    whWHQJ//r'   c                 N    t          |          }t          | |dz   z             S rG   )r   r   r4   r$   rv  rK   s       r%   rV   zplanck_gen._cdf  s(    !HHwh!n%%%%r'   c                 H    t          |                     ||                    S r    )r   r   )r4   r$   rv  s      r%   rZ   zplanck_gen._sf  s    4;;q'**+++r'   c                 2    t          |          }| |dz   z  S rG   r   ry  s       r%   r   zplanck_gen._logsf  s    !HHx1~r'   c                     t          d|z  t          |           z  dz
            } |dz
  j        |                     |           }|                     ||          }t          j        ||k    ||          S )N      r   )r   r   cliprE   rV   r!   r   )r4   rb   rv  rw   rj  r   s         r%   rc   zplanck_gen._ppf  sp    DL5!99,Q.//a 1 1' : :<yy((x	5$///r'   Nc                 X    t          |            }|                    ||          dz
  S )Nr   r   )r   r   )r4   rv  r:   r;   r.   s        r%   r<   zplanck_gen._rvs  s0    G8__%%ad%33c99r'   c                     dt          |          z  }t          |           t          |           dz  z  }dt          |dz            z  }ddt          |          z  z   }||||fS )Nr   r   r   rR  )r   r   r   )r4   rv  rm   rn   ro   rp   s         r%   rq   zplanck_gen._stats  si    uW~~7(mmUG8__q00tGCK   qg3Br'   c                 p    t          |            }|t          |           z  |z  t          |          z
  S r    )r   r   r   )r4   rv  Cs      r%   rx   zplanck_gen._entropy  s5    G8__sG8}}$Q&Q//r'   rf   )r{   r|   r}   r~   r5   rA   rR   rV   rZ   r   rc   r<   rq   rx   r   r'   r%   rr  rr  }  s         6J J J  0 0 0& & &, , ,  0 0 0: : : :
  0 0 0 0 0r'   rr  planckzA discrete exponential c                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
boltzmann_gena  A Boltzmann (Truncated Discrete Exponential) random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `boltzmann` is:

    .. math::

        f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) / (1-\exp(-\lambda N))

    for :math:`k = 0,..., N-1`.

    `boltzmann` takes :math:`\lambda > 0` and :math:`N > 0` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 z    t          dddt          j        fd          t          dddt          j        fd          gS )Nrv  Fr   r   r   Tr1   r3   s    r%   r5   zboltzmann_gen._shape_info  s<    9ea[.II3q"&k>BBD 	Dr'   c                 <    |dk    |dk    z  t          |          z  S r   r?   r4   rv  r   s      r%   rA   zboltzmann_gen._argcheck  s     !A&Q77r'   c                     | j         |dz
  fS rG   rC   r  s      r%   rE   zboltzmann_gen._get_support  s    vq1u}r'   c                     dt          |           z
  dt          | |z            z
  z  }|t          | |z            z  S rG   r   )r4   rK   rv  r   facts        r%   rR   zboltzmann_gen._pmf  sB     #wh--!C
OO"34C
OO##r'   c                     t          |          }dt          | |dz   z            z
  dt          | |z            z
  z  S rG   )r   r   )r4   r$   rv  r   rK   s        r%   rV   zboltzmann_gen._cdf  s@    !HH#wh!n%%%#whqj//(9::r'   c                 ,   |dt          | |z            z
  z  }t          d|z  t          d|z
            z  dz
            }|dz
                      dt          j                  }|                     |||          }t	          j        ||k    ||          S )Nr   r~  rD  )r   r   r   r  r!   r2   rV   r   )r4   rb   rv  r   qnewrw   rj  r   s           r%   rc   zboltzmann_gen._ppf  s    !C
OO#$DL3qv;;.q011ac26**yy++x	5$///r'   c                    t          |           }t          | |z            }|d|z
  z  ||z  d|z
  z  z
  }|d|z
  dz  z  ||z  |z  d|z
  dz  z  z
  }d|z
  d|z
  z  }||dz  z  ||z  |z  z
  }|d|z   z  |dz  z  |dz  |z  d|z   z  z
  }	|	|dz  z  }	|dd|z  z   ||z  z   z  |dz  z  |dz  |z  dd|z  z   ||z  z   z  z
  }
|
|z  |z  }
|||	|
fS )Nr   r   r   r   rQ  rR  r  )r4   rv  r   zzNrm   rn   trmtrm2ro   rp   s              r%   rq   zboltzmann_gen._stats  s,   MM'!__AYqtQrT{"Q
lQqSVQrTAI--tacl#q&1Q3r6!!WS!V^ad2gqtn,$+!A#ac	]36!AqD2Iq2vbe|$<<$Y3Br'   N)r{   r|   r}   r~   r5   rA   rE   rR   rV   rc   rq   r   r'   r%   r  r    s         *D D D8 8 8  $ $ $; ; ;0 0 0    r'   r  	boltzmannz!A truncated discrete exponential )r   rD   r   c                   J    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d	S )randint_gena  A uniform discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `randint` is:

    .. math::

        f(k) = \frac{1}{\texttt{high} - \texttt{low}}

    for :math:`k \in \{\texttt{low}, \dots, \texttt{high} - 1\}`.

    `randint` takes :math:`\texttt{low}` and :math:`\texttt{high}` as shape
    parameters.

    %(after_notes)s

    %(example)s

    c                     t          ddt          j         t          j        fd          t          ddt          j         t          j        fd          gS )NlowTr   highr1   r3   s    r%   r5   zrandint_gen._shape_info"  sF    5$"&"&(9>JJ6426'26):NKKM 	Mr'   c                 N    ||k    t          |          z  t          |          z  S r    r?   r4   r  r  s      r%   rA   zrandint_gen._argcheck&  s&    s
k#...T1B1BBBr'   c                     ||dz
  fS rG   r   r  s      r%   rE   zrandint_gen._get_support)  s    DF{r'   c                 x    t          j        |          ||z
  z  }t          j        ||k    ||k     z  |d          S )NrD  )r!   	ones_liker   )r4   rK   r  r  r.   s        r%   rR   zrandint_gen._pmf,  s9    LOOtcz*xca$h/B777r'   c                 <    t          |          }||z
  dz   ||z
  z  S r   r|  )r4   r$   r  r  rK   s        r%   rV   zrandint_gen._cdf1  s$    !HHC",,r'   c                     t          |||z
  z  |z             dz
  }|dz
                      ||          }|                     |||          }t          j        ||k    ||          S rG   )r   r  rV   r!   r   )r4   rb   r  r  rw   rj  r   s          r%   rc   zrandint_gen._ppf5  sf    A$s*++a/T**yyT**x	5$///r'   c                     t          j        |          t          j        |          }}||z   dz
  dz  }||z
  }||z  dz
  dz  }d}d||z  dz   z  ||z  dz
  z  }	||||	fS )Nr   r   r   g      (@rD  g333333)r!   r"  )
r4   r  r  m2m1rm   drn   ro   rp   s
             r%   rq   zrandint_gen._stats;  s|    D!!2:c??B2gmq GsQw$1s#qsSy13Br'   Nc                 t   t          j        |          j        dk    r0t          j        |          j        dk    rt          ||||          S |*t          j        ||          }t          j        ||          }t          j        t          t          |          t           j        g          } |||          S )z=An array of *size* random integers >= ``low`` and < ``high``.r   r   N)otypes)r!   r"  r:   r   broadcast_to	vectorizer   int_)r4   r  r  r:   r;   randints         r%   r<   zrandint_gen._rvsD  s    :c??1$$D)9)9)>!)C)Cc4dCCCC
 /#t,,C?4..D,w|\BB')wi1 1 1wsD!!!r'   c                 &    t          ||z
            S r    )r   r  s      r%   rx   zrandint_gen._entropyU  s    4#:r'   rf   )r{   r|   r}   r~   r5   rA   rE   rR   rV   rc   rq   r<   rx   r   r'   r%   r  r  
  s         .M M MC C C  8 8 8
- - -0 0 0  " " " ""    r'   r  r  z#A discrete uniform (random integer)c                   2    e Zd ZdZd ZddZd Zd Zd ZdS )	zipf_gena  A Zipf (Zeta) discrete random variable.

    %(before_notes)s

    See Also
    --------
    zipfian

    Notes
    -----
    The probability mass function for `zipf` is:

    .. math::

        f(k, a) = \frac{1}{\zeta(a) k^a}

    for :math:`k \ge 1`, :math:`a > 1`.

    `zipf` takes :math:`a > 1` as shape parameter. :math:`\zeta` is the
    Riemann zeta function (`scipy.special.zeta`)

    The Zipf distribution is also known as the zeta distribution, which is
    a special case of the Zipfian distribution (`zipfian`).

    %(after_notes)s

    References
    ----------
    .. [1] "Zeta Distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Zeta_distribution

    %(example)s

    Confirm that `zipf` is the large `n` limit of `zipfian`.

    >>> import numpy as np
    >>> from scipy.stats import zipfian
    >>> k = np.arange(11)
    >>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000))
    True

    c                 @    t          dddt          j        fd          gS )NrD   Fr   r   r1   r3   s    r%   r5   zzipf_gen._shape_info      326{NCCDDr'   Nc                 0    |                     ||          S r   )zipf)r4   rD   r:   r;   s       r%   r<   zzipf_gen._rvs  s       ...r'   c                     |dk    S rG   r   r4   rD   s     r%   rA   zzipf_gen._argcheck  s    1ur'   c                 B    dt          j        |d          z  ||z  z  }|S Nr   r   r   r	   )r4   rK   rD   ra  s       r%   rR   zzipf_gen._pmf  s&    7<1%%%1,	r'   c                 N    t          ||dz   k    ||fd t          j                  S )Nr   c                 ^    t          j        | |z
  d          t          j        | d          z  S rG   r  )rD   r,   s     r%   rC  z zipf_gen._munp.<locals>.<lambda>  s'    a!eQ//',q!2D2DD r'   )r
   r!   r2   )r4   r,   rD   s      r%   _munpzzipf_gen._munp  s0    AI1vDDF  	r'   rf   )	r{   r|   r}   r~   r5   r<   rA   rR   r  r   r'   r%   r  r  ^  sr        ) )VE E E/ / / /    
    r'   r  r  zA Zipfc                 J    t          |d          t          || dz             z
  S )z"Generalized harmonic number, a > 1r   )r	   r,   rD   s     r%   _gen_harmonic_gt1r    s#     1::Q!$$r'   c                    t          j        |           s| S t          j        |           }t          j        |t                    }t          j        |ddt                    D ]$}|| k    }||xx         d|||         z  z  z  cc<   %|S )z#Generalized harmonic number, a <= 1r   r   r   )r!   r:   max
zeros_likefloatr!  )r,   rD   n_maxoutimasks         r%   _gen_harmonic_leq1r    s    71:: F1IIE
-
'
'
'CYua5111 " "AvD			Qq!D'z\!				Jr'   c                 x    t          j        | |          \  } }t          |dk    | |ft          t                    S )zGeneralized harmonic numberr   ff2)r!   r  r
   r  r  r  s     r%   _gen_harmonicr    sE    q!$$DAqa!eaV).@B B B Br'   c                   <    e Zd ZdZd Zd Zd Zd Zd Zd Z	d Z
d	S )
zipfian_gena{  A Zipfian discrete random variable.

    %(before_notes)s

    See Also
    --------
    zipf

    Notes
    -----
    The probability mass function for `zipfian` is:

    .. math::

        f(k, a, n) = \frac{1}{H_{n,a} k^a}

    for :math:`k \in \{1, 2, \dots, n-1, n\}`, :math:`a \ge 0`,
    :math:`n \in \{1, 2, 3, \dots\}`.

    `zipfian` takes :math:`a` and :math:`n` as shape parameters.
    :math:`H_{n,a}` is the :math:`n`:sup:`th` generalized harmonic
    number of order :math:`a`.

    The Zipfian distribution reduces to the Zipf (zeta) distribution as
    :math:`n \rightarrow \infty`.

    %(after_notes)s

    References
    ----------
    .. [1] "Zipf's Law", Wikipedia, https://en.wikipedia.org/wiki/Zipf's_law
    .. [2] Larry Leemis, "Zipf Distribution", Univariate Distribution
           Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf

    %(example)s

    Confirm that `zipfian` reduces to `zipf` for large `n`, `a > 1`.

    >>> import numpy as np
    >>> from scipy.stats import zipf
    >>> k = np.arange(11)
    >>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
    True

    c                 z    t          dddt          j        fd          t          dddt          j        fd          gS )NrD   Fr   r-   r,   Tr   r1   r3   s    r%   r5   zzipfian_gen._shape_info  s<    326{MBB3q"&k>BBD 	Dr'   c                 \    |dk    |dk    z  |t          j        |t                    k    z  S )Nr   r  )r!   r"  r9  r4   rD   r,   s      r%   rA   zzipfian_gen._argcheck  s.    Q1q5!Q"*Qc*B*B*B%BCCr'   c                 
    d|fS rG   r   r  s      r%   rE   zzipfian_gen._get_support  r   r'   c                 4    dt          ||          z  ||z  z  S r   r  r4   rK   rD   r,   s       r%   rR   zzipfian_gen._pmf  s     ]1a(((1a4//r'   c                 D    t          ||          t          ||          z  S r    r  r  s       r%   rV   zzipfian_gen._cdf  s!    Q""]1a%8%888r'   c                     |dz   }||z  t          ||          t          ||          z
  z  dz   ||z  t          ||          z  z  S rG   r  r  s       r%   rZ   zzipfian_gen._sf  sU    EA}Q**]1a-@-@@AAEa4a+++- 	.r'   c                    t          ||          }t          ||dz
            }t          ||dz
            }t          ||dz
            }t          ||dz
            }||z  }||z  |dz  z
  }	|dz  }
|	|
z  }||z  d|z  |z  |dz  z  z
  d|dz  z  |dz  z  z   |dz  z  }|dz  |z  d|dz  z  |z  |z  z
  d|z  |dz  z  |z  z   d|dz  z  z
  |	dz  z  }|dz  }||||fS )Nr   r   r   rR  rQ  r   r  )r4   rD   r,   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2ro   rp   s                 r%   rq   zzipfian_gen._stats  s5   Aq!!Q!$$Q!$$Q!$$Q!$$3hS47"AvTk3h4S!V++aaiQ.>>c
J1fTkAc1fHTM$..3tQwt1CC$'	!1W%
aCRr'   N)r{   r|   r}   r~   r5   rA   rE   rR   rV   rZ   rq   r   r'   r%   r  r    s        , ,\D D DD D D  0 0 09 9 9. . .         r'   r  zipfianz	A Zipfianc                   >    e Zd ZdZd Zd Zd Zd Zd Zd Z	d
d	Z
dS )dlaplace_genaL  A  Laplacian discrete random variable.

    %(before_notes)s

    Notes
    -----
    The probability mass function for `dlaplace` is:

    .. math::

        f(k) = \tanh(a/2) \exp(-a |k|)

    for integers :math:`k` and :math:`a > 0`.

    `dlaplace` takes :math:`a` as shape parameter.

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS )NrD   Fr   r   r1   r3   s    r%   r5   zdlaplace_gen._shape_info.  r  r'   c                 h    t          |dz            t          | t          |          z            z  S Nr   )r   r   abs)r4   rK   rD   s      r%   rR   zdlaplace_gen._pmf1  s+    AcE{{S!c!ff----r'   c                 ^    t          |          }d }d }t          |dk    ||f||          S )Nc                 T    dt          | | z            t          |          dz   z  z
  S r  r  rK   rD   s     r%   r  zdlaplace_gen._cdf.<locals>.f8  s(    aR!VA
333r'   c                 R    t          || dz   z            t          |          dz   z  S rG   r  r  s     r%   r  zdlaplace_gen._cdf.<locals>.f2;  s'    qAE{##s1vvz22r'   r   r  )r   r
   )r4   r$   rD   rK   r  r  s         r%   rV   zdlaplace_gen._cdf5  sL    !HH	4 	4 	4	3 	3 	3 !q&1a&A"5555r'   c           
      \   dt          |          z   }t          t          j        |ddt          |           z   z  k     t	          ||z            |z  dz
  t	          d|z
  |z             |z                      }|dz
  }t          j        |                     ||          |k    ||          S )Nr   r   )r   r   r!   r   r   rV   )r4   rb   rD   constrw   rj  s         r%   rc   zdlaplace_gen._ppf@  s    CFF
BHQCGG!44 5\\A-1!1Q3%-000146 6 7 7 qx		%++q0%>>>r'   c                     t          |          }d|z  |dz
  dz  z  }d|z  |dz  d|z  z   dz   z  |dz
  dz  z  }d|d||dz  z  dz
  fS )Nr   r   r   g      $@rR  rD  r  r  )r4   rD   ear  rW  s        r%   rq   zdlaplace_gen._statsH  si    VVeRUQJeRU3r6\"_%B
23CQJO++r'   c                 f    |t          |          z  t          t          |dz                      z
  S r  )r   r   r   r  s     r%   rx   zdlaplace_gen._entropyN  s)    477{Sae----r'   Nc                     t          j        t          j        |                      }|                    ||          }|                    ||          }||z
  S r   )r!   r   r"  r   )r4   rD   r:   r;   probOfSuccessr$   ys          r%   r<   zdlaplace_gen._rvsQ  sY      2:a==.111""=t"<<""=t"<<1ur'   rf   )r{   r|   r}   r~   r5   rR   rV   rc   rq   rx   r<   r   r'   r%   r  r    s         ,E E E. . .	6 	6 	6? ? ?, , ,. . .     r'   r  dlaplacezA discrete Laplacianc                   2    e Zd ZdZd ZddZd Zd Zd ZdS )	skellam_gena  A  Skellam discrete random variable.

    %(before_notes)s

    Notes
    -----
    Probability distribution of the difference of two correlated or
    uncorrelated Poisson random variables.

    Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with
    expected values :math:`\lambda_1` and :math:`\lambda_2`. Then,
    :math:`k_1 - k_2` follows a Skellam distribution with parameters
    :math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and
    :math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where
    :math:`\rho` is the correlation coefficient between :math:`k_1` and
    :math:`k_2`. If the two Poisson-distributed r.v. are independent then
    :math:`\rho = 0`.

    Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.

    For details see: https://en.wikipedia.org/wiki/Skellam_distribution

    `skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 z    t          dddt          j        fd          t          dddt          j        fd          gS )Nr  Fr   r   r  r1   r3   s    r%   r5   zskellam_gen._shape_info  s<    5%!RVnEE5%!RVnEEG 	Gr'   Nc                 `    |}|                     ||          |                     ||          z
  S r    r^  )r4   r  r  r:   r;   r,   s         r%   r<   zskellam_gen._rvs  s7    $$S!,,$$S!,,- 	.r'   c                     t          j        d          5  t          j        |dk     t          j        d|z  dd|z
  z  d|z            dz  t          j        d|z  dd|z   z  d|z            dz            }d d d            n# 1 swxY w Y   |S )Nr   r   r   r   r   )r!   r   r   rO   	_ncx2_pdfr4   r$   r  r  pxs        r%   rR   zskellam_gen._pmf  s    [h''' 	E 	E!a% *1S5!QqS'1S5AA!C *1S5!QqS'1S5AA!CE EB	E 	E 	E 	E 	E 	E 	E 	E 	E 	E 	E 	E 	E 	E 	E
 	s   A!BB
Bc                 2   t          |          }t          j        d          5  t          j        |dk     t	          j        d|z  d|z  d|z            dt	          j        d|z  d|dz   z  d|z            z
            }d d d            n# 1 swxY w Y   |S )Nr   r   r   r   r   )r   r!   r   r   rO   	_ncx2_cdfr  s        r%   rV   zskellam_gen._cdf  s    !HH[h''' 	G 	G!a% *1S5"Q$#>>f.qua1gquEEEG GB	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	G 	s   ABBBc                 V    ||z
  }||z   }|t          |dz            z  }d|z  }||||fS )Nr   r   rm  )r4   r  r  meanrn   ro   rp   s          r%   rq   zskellam_gen._stats  s@    SyCiD#NN"WS"b  r'   rf   )	r{   r|   r}   r~   r5   r<   rR   rV   rq   r   r'   r%   r  r  k  sq         :G G G. . . .
    ! ! ! ! !r'   r  skellamz	A Skellamc                   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 )yulesimon_gena  A Yule-Simon discrete random variable.

    %(before_notes)s

    Notes
    -----

    The probability mass function for the `yulesimon` is:

    .. math::

        f(k) =  \alpha B(k, \alpha+1)

    for :math:`k=1,2,3,...`, where :math:`\alpha>0`.
    Here :math:`B` refers to the `scipy.special.beta` function.

    The sampling of random variates is based on pg 553, Section 6.3 of [1]_.
    Our notation maps to the referenced logic via :math:`\alpha=a-1`.

    For details see the wikipedia entry [2]_.

    References
    ----------
    .. [1] Devroye, Luc. "Non-uniform Random Variate Generation",
         (1986) Springer, New York.

    .. [2] https://en.wikipedia.org/wiki/Yule-Simon_distribution

    %(after_notes)s

    %(example)s

    c                 @    t          dddt          j        fd          gS )NalphaFr   r   r1   r3   s    r%   r5   zyulesimon_gen._shape_info  s    7EArv;GGHHr'   Nc           	          |                     |          }|                     |          }t          | t          t          | |z                       z            }|S r    )standard_exponentialr   r   r   )r4   r	  r:   r;   E1E2anss          r%   r<   zyulesimon_gen._rvs  sZ    ..t44..t44B3RC%K 0 0011122
r'   c                 8    |t          j        ||dz             z  S rG   r   r   r4   r$   r	  s      r%   rR   zyulesimon_gen._pmf  s    w|Auqy1111r'   c                     |dk    S r   r   )r4   r	  s     r%   rA   zyulesimon_gen._argcheck  s    	r'   c                 R    t          |          t          j        ||dz             z   S rG   r   r   r   r  s      r%   rM   zyulesimon_gen._logpmf  s#    5zzGN1eai8888r'   c                 >    d|t          j        ||dz             z  z
  S rG   r  r  s      r%   rV   zyulesimon_gen._cdf  s"    1w|Auqy11111r'   c                 8    |t          j        ||dz             z  S rG   r  r  s      r%   rZ   zyulesimon_gen._sf  s    7<519----r'   c                 R    t          |          t          j        ||dz             z   S rG   r  r  s      r%   r   zyulesimon_gen._logsf  s#    1vvq%!)4444r'   c                    t          j        |dk    t           j        ||dz
  z            }t          j        |dk    |dz  |dz
  |dz
  dz  z  z  t           j                  }t          j        |dk    t           j        |          }t          j        |dk    t	          |dz
            |dz   dz  z  ||dz
  z  z  t           j                  }t          j        |dk    t           j        |          }t          j        |dk    |dz   |dz  d|z  z
  dz
  ||dz
  z  |dz
  z  z  z   t           j                  }t          j        |dk    t           j        |          }||||fS )Nr   r   r   r   rR  1      )r!   r   r2   nanr   )r4   r	  rm   r  ro   rp   s         r%   rq   zyulesimon_gen._stats  sX   Xeqj"&%519*=>>huqyaxECKEAI>#ABv  huz263//Xeai519ooQ6%519:MNf  Xeqj"&"--XeaiaiE1HrEz$9B$>$)UQY$7519$E$G Hf  Xeqj"&"--3Br'   rf   )r{   r|   r}   r~   r5   r<   rR   rA   rM   rV   rZ   r   rq   r   r'   r%   r  r    s           BI I I   2 2 2  9 9 92 2 2. . .5 5 5    r'   r  	yulesimon)r   rD   c                       fd}|S )z?Decorator that vectorizes _rvs method to work on ndarray shapesc                 X   t          |d         j        |           \  }}t          j        |           } t          j        |          }t          j        |          }t          j        |          r 	g || |R  S t          j        |           }t          j        |j                  }t          j        ||          ||         f          }t          j	        |||          }t          j
        | |           D ] 	g fd|D             ||R  |<   t          j	        |||          S )Nr   c                 D    g | ]}t          j        |                   S r   )r!   squeeze).0argr  s     r%   
<listcomp>z<_vectorize_rvs_over_shapes.<locals>._rvs.<locals>.<listcomp>  s&    @@@CRZ__Q/@@@r'   )r   shaper!   arrayallemptyr!  ndimhstackmoveaxisndindex)
r:   r;   args
_rvs1_size_rvs1_indicesr  j0j1r  r>  s
           @r%   r<   z(_vectorize_rvs_over_shapes.<locals>._rvs  sK   $0a$E$E!
Mx~~Xj))
//6-   	453$33l3333htnn
 Ysx  YM>*B},=>??k#r2&&T=.12 	5 	5A U 5@@@@4@@@ 5%5'35 5 5CFF {3B'''r'   r   )r>  r<   s   ` r%   r@  r@    s#    ( ( ( ( (4 Kr'   c                   @    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 )
_nchypergeom_genzA noncentral hypergeometric discrete random variable.

    For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.

    Nc           	          t          dddt          j        fd          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   Tr   r-   r,   r   oddsFr   r1   r3   s    r%   r5   z_nchypergeom_gen._shape_info/  sj    3q"&k=AA3q"&k=AA3q"&k=AA651bf+~FFH 	Hr'   c                 z    |||}}}||z
  }t          j        d||z
            }t          j        ||          }||fS r   r  )	r4   r   r,   r   r4  r  r  x_minx_maxs	            r%   rE   z_nchypergeom_gen._get_support5  sH    aq2V
1a"f%%
1b!!e|r'   c                    t          j        |          t          j        |          }}t          j        |          t          j        |          }}|                    t                    |k    |dk    z  }|                    t                    |k    |dk    z  }|                    t                    |k    |dk    z  }|dk    }||k    }	||k    }
||z  |z  |z  |	z  |
z  S r   )r!   r"  r   r9  )r4   r   r,   r   r4  cond1cond2cond3cond4cond5cond6s              r%   rA   z_nchypergeom_gen._argcheck<  s    z!}}bjmm1*Q--D!1!14#!#Q/#!#Q/#!#Q/qQQu}u$u,u4u<<r'   c                 J     t            fd            } |||||||          S )Nc                     t          j        |          }t                      }t          |
j                  } |||| |||          }	|	                    |          }	|	S r    )r!   prodr   getattrrvs_namereshape)r   r,   r   r4  r:   r;   lengthurnrv_genr=  r4   s             r%   r>  z$_nchypergeom_gen._rvs.<locals>._rvs1I  s[    WT]]F#%%CS$-00F&Aq$==C++d##CJr'   r   r?  )r4   r   r,   r   r4  r:   r;   r>  s   `       r%   r<   z_nchypergeom_gen._rvsG  sF    	#	 	 	 	 
$	#	 uQ1dLIIIIr'   c                      t          j        |||||          \  }}}}}|j        dk    rt          j        |          S t           j         fd            } ||||||          S )Nr   c                 `                         ||||d          }|                    |           S Ng-q=)distprobability)r$   r   r,   r   r4  rF  r4   s         r%   _pmf1z$_nchypergeom_gen._pmf.<locals>._pmf1Z  s.    ))Aq!T511C??1%%%r'   )r!   r  r:   
empty_liker  )r4   r$   r   r,   r   r4  rM  s   `      r%   rR   z_nchypergeom_gen._pmfT  s    .q!Q4@@1aD6Q;;=###		& 	& 	& 	& 
	& uQ1a&&&r'   c                 ~     t           j         fd            }d|v sd|v r |||||          nd\  }}d\  }	}
|||	|
fS )Nc                 ^                         ||| |d          }|                                S rJ  )rK  rl   )r   r,   r   r4  rF  r4   s        r%   	_moments1z*_nchypergeom_gen._stats.<locals>._moments1c  s*    ))Aq!T511C;;== r'   r  vrf   )r!   r  )r4   r   r,   r   r4  rl   rQ  r  rR  rg   rK   s   `          r%   rq   z_nchypergeom_gen._statsa  ss    		! 	! 	! 	! 
	! .1G^^sg~~		!Q4(((! 	11!Qzr'   rf   )r{   r|   r}   r~   rC  rK  r5   rE   rA   r<   rR   rq   r   r'   r%   r2  r2  %  s          HDH H H  	= 	= 	=J J J J' ' '
 
 
 
 
r'   r2  c                       e Zd ZdZdZeZdS )nchypergeom_fisher_genag	  A Fisher's noncentral hypergeometric discrete random variable.

    Fisher's noncentral hypergeometric distribution models drawing objects of
    two types from a bin. `M` is the total number of objects, `n` is the
    number of Type I objects, and `odds` is the odds ratio: the odds of
    selecting a Type I object rather than a Type II object when there is only
    one object of each type.
    The random variate represents the number of Type I objects drawn if we
    take a handful of objects from the bin at once and find out afterwards
    that we took `N` objects.

    %(before_notes)s

    See Also
    --------
    nchypergeom_wallenius, hypergeom, nhypergeom

    Notes
    -----
    Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
    with parameters `N`, `n`, and `M` (respectively) as defined above.

    The probability mass function is defined as

    .. math::

        p(x; M, n, N, \omega) =
        \frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},

    for
    :math:`x \in [x_l, x_u]`,
    :math:`M \in {\mathbb N}`,
    :math:`n \in [0, M]`,
    :math:`N \in [0, M]`,
    :math:`\omega > 0`,
    where
    :math:`x_l = \max(0, N - (M - n))`,
    :math:`x_u = \min(N, n)`,

    .. math::

        P_0 = \sum_{y=x_l}^{x_u} \binom{n}{y}\binom{M - n}{N-y}\omega^y,

    and the binomial coefficients are defined as

    .. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.

    `nchypergeom_fisher` uses the BiasedUrn package by Agner Fog with
    permission for it to be distributed under SciPy's license.

    The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
    universally accepted; they are chosen for consistency with `hypergeom`.

    Note that Fisher's noncentral hypergeometric distribution is distinct
    from Wallenius' noncentral hypergeometric distribution, which models
    drawing a pre-determined `N` objects from a bin one by one.
    When the odds ratio is unity, however, both distributions reduce to the
    ordinary hypergeometric distribution.

    %(after_notes)s

    References
    ----------
    .. [1] Agner Fog, "Biased Urn Theory".
           https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf

    .. [2] "Fisher's noncentral hypergeometric distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Fisher's_noncentral_hypergeometric_distribution

    %(example)s

    
rvs_fisherN)r{   r|   r}   r~   rC  r   rK  r   r'   r%   rT  rT  n  s'        G GR H%DDDr'   rT  nchypergeom_fisherz$A Fisher's noncentral hypergeometricc                       e Zd ZdZdZeZdS )nchypergeom_wallenius_gena}	  A Wallenius' noncentral hypergeometric discrete random variable.

    Wallenius' noncentral hypergeometric distribution models drawing objects of
    two types from a bin. `M` is the total number of objects, `n` is the
    number of Type I objects, and `odds` is the odds ratio: the odds of
    selecting a Type I object rather than a Type II object when there is only
    one object of each type.
    The random variate represents the number of Type I objects drawn if we
    draw a pre-determined `N` objects from a bin one by one.

    %(before_notes)s

    See Also
    --------
    nchypergeom_fisher, hypergeom, nhypergeom

    Notes
    -----
    Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
    with parameters `N`, `n`, and `M` (respectively) as defined above.

    The probability mass function is defined as

    .. math::

        p(x; N, n, M) = \binom{n}{x} \binom{M - n}{N-x}
        \int_0^1 \left(1-t^{\omega/D}\right)^x\left(1-t^{1/D}\right)^{N-x} dt

    for
    :math:`x \in [x_l, x_u]`,
    :math:`M \in {\mathbb N}`,
    :math:`n \in [0, M]`,
    :math:`N \in [0, M]`,
    :math:`\omega > 0`,
    where
    :math:`x_l = \max(0, N - (M - n))`,
    :math:`x_u = \min(N, n)`,

    .. math::

        D = \omega(n - x) + ((M - n)-(N-x)),

    and the binomial coefficients are defined as

    .. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.

    `nchypergeom_wallenius` uses the BiasedUrn package by Agner Fog with
    permission for it to be distributed under SciPy's license.

    The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
    universally accepted; they are chosen for consistency with `hypergeom`.

    Note that Wallenius' noncentral hypergeometric distribution is distinct
    from Fisher's noncentral hypergeometric distribution, which models
    take a handful of objects from the bin at once, finding out afterwards
    that `N` objects were taken.
    When the odds ratio is unity, however, both distributions reduce to the
    ordinary hypergeometric distribution.

    %(after_notes)s

    References
    ----------
    .. [1] Agner Fog, "Biased Urn Theory".
           https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf

    .. [2] "Wallenius' noncentral hypergeometric distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Wallenius'_noncentral_hypergeometric_distribution

    %(example)s

    rvs_walleniusN)r{   r|   r}   r~   rC  r   rK  r   r'   r%   rX  rX    s'        G GR H'DDDr'   rX  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)]	functoolsr   scipyr   scipy.specialr   r   r   r   rH   r	   scipy._lib._utilr
   r   scipy.interpolater   numpyr   r   r   r   r   r   r   r   r   r   r!   _distn_infrastructurer   r   r   r   scipy.stats._booststatsrO   
_biasedurnr   r   r   r&   r)   r   r   r   r   r   r   r   r   r   r   r*  r,  rH  rJ  rX  rZ  r_  rr  r  r  r  r  r  r  r  r  r  r  r  r  r  r2   r  r  r  r  r  r@  r2  rT  rV  rX  rZ  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r   r'   r%   <module>rm     s  
             I I I I I I I I I I I I I I 5 5 5 5 5 5 5 5 & & & & & & M M M M M M M M M M M M M M M M M M M M M M M M    > > > > > > > > > > > > # # # # # # # # #, , , , , , , , , ,
  M* M* M* M* M* M* M* M*` 		w># ># ># ># >#I ># ># >#B MAK000	O O O O OK O O Od M{+++	j
 j
 j
 j
 j
 j
 j
 j
Z 
	"	"	"E7 E7 E7 E7 E7{ E7 E7 E7P x!&=999Q Q Q Q QK Q Q Qh M{+++	\ \ \ \ \[ \ \ \~ ^...
5 5 5 5 5 5 5 5p 
ah	A	A	A? ? ? ? ?+ ? ? ?D +9{
;
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;D0 D0 D0 D0 D0 D0 D0 D0N 
ah1J	K	K	K< < < < <K < < <~ M{a#FH H H	L L L L L+ L L L^ +9 0) * * *
> > > > >{ > > >B x!&8444% % %
 
 
B B BU  U  U  U  U + U  U  U p +	K
@
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@M M M M M; M M M` <26''2HJ J J<! <! <! <! <!+ <! <! <!~ +i+
F
F
FL L L L LK L L L^ M{a000	# # #LF F F F F{ F F FRK& K& K& K& K&- K& K& K&\ ,+	35 5 5 
K( K( K( K( K( 0 K( K( K(\ 21	 57 7 7  	WWYY^^##%%&&!7!7{!K!K 
)
)r'   