o
    5ήc                     @   s\  d dl mZ d dl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!m"Z"m#Z# d dl$m%  m&Z& dd
l'm(Z(m)Z)m*Z* dd Z+G dd deZ,e,ddZ-G dd de,Z.e.dddZ/G dd deZ0e0ddZ1G dd deZ2e2ddZ3G dd deZ4e4ddddZ5G d d! d!eZ6e6d"dZ7G d#d$ d$eZ8e8d%dZ9G d&d' d'eZ:e:dd(d)dZ;G d*d+ d+eZ<e<d,d-d.Z=G d/d0 d0eZ>e>d d1d2dZ?G d3d4 d4eZ@e@d5d d6d7ZAG d8d9 d9eZBeBd:d;d.ZCG d<d= d=eZDeDdd>d?dZEd@dA ZFdBdC ZGdDdE ZHG dFdG dGeZIeIddHdIdZJG dJdK dKeZKeKejL dLdMdZMG dNdO dOeZNeNejL dPdQdZOG dRdS dSeZPePdTddUZQdVdW ZRG dXdY dYeZSG dZd[ d[eSZTeTd\d]d.ZUG d^d_ d_eSZVeVd`dad.ZWeXeY Z [ Z\e!e\e\Z]Z^e]e^ Z_dS )b    )partialN)special)entr	logsumexpbetalngammalnzeta)
_lazywhererng_integers)interp1d)
floorceillogexpsqrtlog1pexpm1tanhcoshsinh   )rv_discrete	_ncx2_pdf	_ncx2_cdfget_distribution_names_check_shape
_ShapeInfo)_PyFishersNCHypergeometric_PyWalleniusNCHypergeometric_PyStochasticLib3c                 C   s   | t | kS N)nproundx r%   C/tmp/pip-target-vg8gfxp4/lib/python/scipy/stats/_discrete_distns.py_isintegral      r'   c                   @   st   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	dd Z
dd Zdd Zdd ZdddZd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                 C   "   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selfr%   r%   r&   _shape_info:      zbinom_gen._shape_infoNc                 C      | |||S r    )binomialr4   r,   r.   sizerandom_stater%   r%   r&   _rvs>   r(   zbinom_gen._rvsc                 C   s    |dkt |@ |dk@ |dk@ S Nr   r   r'   r4   r,   r.   r%   r%   r&   	_argcheckA       zbinom_gen._argcheckc                 C   s
   | j |fS r    ar?   r%   r%   r&   _get_supportD   s   
zbinom_gen._get_supportc                 C   sR   t |}t|d t|d t|| d   }|t|| t|| |  S Nr   )r   gamlnr   xlogyxlog1py)r4   r$   r,   r.   kcombilnr%   r%   r&   _logpmfG   s   ("zbinom_gen._logpmfc                 C   s>   t jdd t|||W  d    S 1 sw   Y  d S )Nignoredivide)r!   errstate_boost
_binom_pdfr4   r$   r,   r.   r%   r%   r&   _pmfL   s   $zbinom_gen._pmfc                 C      t |}t|||S r    )r   rP   
_binom_cdfr4   r$   r,   r.   rI   r%   r%   r&   _cdfR      zbinom_gen._cdfc                 C   rT   r    )r   rP   	_binom_sfrV   r%   r%   r&   _sfV   rX   zbinom_gen._sfc                 C      t |||S r    )rP   
_binom_isfrR   r%   r%   r&   _isfZ   r(   zbinom_gen._isfc                 C   r[   r    )rP   
_binom_ppf)r4   qr,   r.   r%   r%   r&   _ppf]   r(   zbinom_gen._ppfmvc                 C   sT   t ||}t ||}d\}}d|v rt ||}d|v r$t ||}||||fS )NNNsrI   )rP   _binom_mean_binom_variance_binom_skewness_binom_kurtosis_excess)r4   r,   r.   momentsmuvarg1g2r%   r%   r&   _stats`   s   zbinom_gen._statsc                 C   s2   t jd|d  }| |||}t jt|ddS )Nr   r   axis)r!   r_rS   sumr   )r4   r,   r.   rI   valsr%   r%   r&   _entropyj   s   zbinom_gen._entropyrb   ra   __name__
__module____qualname____doc__r5   r<   r@   rD   rK   rS   rW   rZ   r]   r`   rm   rs   r%   r%   r%   r&   r)      s    


r)   binom)namec                   @   r   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	dd Z
dd Zdd Zdd Zdd Zd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                 C      t ddddgS Nr.   Fr/   r0   r   r3   r%   r%   r&   r5         zbernoulli_gen._shape_infoNc                 C   s   t j| d|||dS )Nr   r:   r;   )r)   r<   r4   r.   r:   r;   r%   r%   r&   r<         zbernoulli_gen._rvsc                 C   s   |dk|dk@ S r=   r%   r4   r.   r%   r%   r&   r@      r   zbernoulli_gen._argcheckc                 C   s   | j | jfS r    )rC   br   r%   r%   r&   rD      s   zbernoulli_gen._get_supportc                 C      t |d|S rE   )rz   rK   r4   r$   r.   r%   r%   r&   rK      r(   zbernoulli_gen._logpmfc                 C   r   rE   )rz   rS   r   r%   r%   r&   rS         zbernoulli_gen._pmfc                 C   r   rE   )rz   rW   r   r%   r%   r&   rW      r(   zbernoulli_gen._cdfc                 C   r   rE   )rz   rZ   r   r%   r%   r&   rZ      r(   zbernoulli_gen._sfc                 C   r   rE   )rz   r]   r   r%   r%   r&   r]      r(   zbernoulli_gen._isfc                 C   r   rE   )rz   r`   )r4   r_   r.   r%   r%   r&   r`      r(   zbernoulli_gen._ppfc                 C   s   t d|S rE   )rz   rm   r   r%   r%   r&   rm         zbernoulli_gen._statsc                 C   s   t |t d|  S rE   )r   r   r%   r%   r&   rs      r   zbernoulli_gen._entropyrb   ru   r%   r%   r%   r&   r}   s   s    
r}   	bernoulli)r   r{   c                   @   sL   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	d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                 C   s:   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-   rC   FFFr   r1   r3   r%   r%   r&   r5         zbetabinom_gen._shape_infoNc                 C   s   | |||}||||S r    )betar8   )r4   r,   rC   r   r:   r;   r.   r%   r%   r&   r<      s   zbetabinom_gen._rvsc                 C   s   d|fS Nr   r%   r4   r,   rC   r   r%   r%   r&   rD         zbetabinom_gen._get_supportc                 C   s    |dkt |@ |dk@ |dk@ S r   r>   r   r%   r%   r&   r@      rA   zbetabinom_gen._argcheckc                 C   sP   t |}t|d  t|| d |d  }|t|| || |  t|| S rE   )r   r   r   )r4   r$   r,   rC   r   rI   rJ   r%   r%   r&   rK      s   $$zbetabinom_gen._logpmfc                 C      t | ||||S r    r   rK   )r4   r$   r,   rC   r   r%   r%   r&   rS      r   zbetabinom_gen._pmfra   c                 C   sx  |||  }d| }|| }||| |  | | || d  }d\}	}
d|v rHdt | }	|	|| d|  ||  9 }	|	|| d ||   }	d|v r|| }
|
|| d d|  9 }
|
d| | |d  7 }
|
d|d  7 }
|
d| | | d|  8 }
|
d	| | |d  8 }
|
|| d d| |  9 }
|
|| | || d  || d  || |   }
|
d8 }
|||	|
fS )
Nr   rb   rc         ?   rI            r   )r4   r,   rC   r   rh   e_pe_qri   rj   rk   rl   r%   r%   r&   rm      s(   $4zbetabinom_gen._statsrb   rt   )rv   rw   rx   ry   r5   r<   rD   r@   rK   rS   rm   r%   r%   r%   r&   r      s    #
r   	betabinomc                   @   sj   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	dd Z
dd Zdd Zdd Zd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 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                 C   r*   r+   r1   r3   r%   r%   r&   r5   ?  r6   znbinom_gen._shape_infoNc                 C   r7   r    )negative_binomialr9   r%   r%   r&   r<   C  r(   znbinom_gen._rvsc                 C   s   |dk|dk@ |dk@ S r=   r%   r?   r%   r%   r&   r@   F     znbinom_gen._argcheckc                 C   r[   r    )rP   _nbinom_pdfrR   r%   r%   r&   rS   I  s   znbinom_gen._pmfc                 C   s>   t || t |d  t | }||t|  t||  S rE   )rF   r   r   rH   )r4   r$   r,   r.   coeffr%   r%   r&   rK   M  s    znbinom_gen._logpmfc                 C   rT   r    )r   rP   _nbinom_cdfrV   r%   r%   r&   rW   Q  rX   znbinom_gen._cdfc           	      C   s   t |}| |||}|dk}dd }|}tjdd" ||| || || ||< t||  || < W d    |S 1 s@w   Y  |S )N      ?c                 S   s   t t| d |d|  S rE   )r!   r   r   betainc)rI   r,   r.   r%   r%   r&   f1Z  s   znbinom_gen._logcdf.<locals>.f1rL   rM   )r   rW   r!   rO   r   )	r4   r$   r,   r.   rI   cdfcondr   logcdfr%   r%   r&   _logcdfU  s   
znbinom_gen._logcdfc                 C   rT   r    )r   rP   
_nbinom_sfrV   r%   r%   r&   rZ   d  rX   znbinom_gen._sfc                 C   L   t   d}t jd|d t|||W  d    S 1 sw   Y  d S )Nz#overflow encountered in _nbinom_isfrL   message)warningscatch_warningsfilterwarningsrP   _nbinom_isf)r4   r$   r,   r.   r   r%   r%   r&   r]   h  s
   
$znbinom_gen._isfc                 C   r   )Nz#overflow encountered in _nbinom_ppfrL   r   )r   r   r   rP   _nbinom_ppf)r4   r_   r,   r.   r   r%   r%   r&   r`   o  s
   
$znbinom_gen._ppfc                 C   s,   t ||t ||t ||t ||fS r    )rP   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessr?   r%   r%   r&   rm   u  s
   



znbinom_gen._statsrb   )rv   rw   rx   ry   r5   r<   r@   rS   rK   rW   r   rZ   r]   r`   rm   r%   r%   r%   r&   r     s    2
r   nbinomc                   @   sb   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	dd Z
dd Zdd Zd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                 C   r~   r   r   r3   r%   r%   r&   r5     r   zgeom_gen._shape_infoNc                 C      |j ||dS Nr:   )	geometricr   r%   r%   r&   r<     r(   zgeom_gen._rvsc                 C   s   |dk|dk@ S Nr   r   r%   r   r%   r%   r&   r@     r   zgeom_gen._argcheckc                 C   s   t d| |d | S rE   )r!   powerr4   rI   r.   r%   r%   r&   rS     r   zgeom_gen._pmfc                 C   s   t |d | t| S rE   )r   rH   r   r   r%   r%   r&   rK        zgeom_gen._logpmfc                 C   s   t |}tt| |  S r    )r   r   r   r4   r$   r.   rI   r%   r%   r&   rW        zgeom_gen._cdfc                 C   s   t | ||S r    )r!   r   _logsfr   r%   r%   r&   rZ     s   zgeom_gen._sfc                 C   s   t |}|t|  S r    )r   r   r   r%   r%   r&   r     rX   zgeom_gen._logsfc                 C   sF   t t| t|  }| |d |}t||k|dk@ |d |S r   )r   r   rW   r!   where)r4   r_   r.   rr   tempr%   r%   r&   r`     s   zgeom_gen._ppfc                 C   sP   d| }d| }|| | }d| t | }tg d|d|  }||||fS )Nr          @)r   ir   )r   r!   polyval)r4   r.   ri   qrrj   rk   rl   r%   r%   r&   rm     s   zgeom_gen._statsrb   )rv   rw   rx   ry   r5   r<   r@   rS   rK   rW   rZ   r   r`   rm   r%   r%   r%   r&   r     s    
r   geomzA geometric)rC   r{   longnamec                   @   r|   )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
    --------
    >>> 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                 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   r%   r%   r&   r5     r   zhypergeom_gen._shape_infoNc                 C   s   |j ||| ||dS r   )hypergeometric)r4   r   r,   r   r:   r;   r%   r%   r&   r<        zhypergeom_gen._rvsc                 C   s    t |||  dt ||fS r   r!   maximumminimum)r4   r   r,   r   r%   r%   r&   rD     rA   zhypergeom_gen._get_supportc                 C   sL   |dk|dk@ |dk@ }|||k||k@ M }|t |t |@ t |@ M }|S r   r>   )r4   r   r,   r   r   r%   r%   r&   r@     s   zhypergeom_gen._argcheckc           	      C   s   ||}}|| }t |d dt |d d t || d |d  t |d || d  t || d || | d  t |d d }|S rE   r   )	r4   rI   r   r,   r   totgoodbadresultr%   r%   r&   rK     s   
0zhypergeom_gen._logpmfc                 C      t ||||S r    )rP   _hypergeom_pdfr4   rI   r   r,   r   r%   r%   r&   rS   $  r   zhypergeom_gen._pmfc                 C   r   r    )rP   _hypergeom_cdfr   r%   r%   r&   rW   '  r   zhypergeom_gen._cdfc                 C   s   d| d| d| }}}|| }||d  d| ||   d| |  }||d | | 9 }|d| | ||  | d| d  7 }||| ||  | |d  |d   }t |||t |||t ||||fS )Nr   r   g      @g      @r   r         @)rP   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r4   r   r,   r   mrl   r%   r%   r&   rm   *  s   (((zhypergeom_gen._statsc                 C   sB   t j|||  t||d  }| ||||}t jt|ddS )Nr   r   rn   )r!   rp   minpmfrq   r   )r4   r   r,   r   rI   rr   r%   r%   r&   rs   ;  s    zhypergeom_gen._entropyc                 C   r   r    )rP   _hypergeom_sfr   r%   r%   r&   rZ   @  r   zhypergeom_gen._sfc                 C   s   g }t t|||| D ]>\}}}}	|d |d  |d |	d  k r3|tt| ||||	  qt|d |	d }
|t| 	|
|||	 qt
|S )Nr   r   )zipr!   broadcast_arraysappendr   r   r   aranger   rK   asarrayr4   rI   r   r,   r   resquantr   r   drawk2r%   r%   r&   r   C  s     "
zhypergeom_gen._logsfc                 C   s   g }t t|||| D ]<\}}}}	|d |d  |d |	d  kr3|tt| ||||	  qtd|d }
|t| 	|
|||	 qt
|S )Nr   r   r   )r   r!   r   r   r   r   logsfr   r   rK   r   r   r%   r%   r&   r   O  s     "
zhypergeom_gen._logcdfrb   )rv   rw   rx   ry   r5   r<   rD   r@   rK   rS   rW   rm   rs   rZ   r   r   r%   r%   r%   r&   r     s    A
r   	hypergeomc                   @   sJ   e Zd ZdZdd Zdd Zdd Zdd	d
Zdd Zdd Z	dd Z
dS )nhypergeom_genaG  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
    --------
    >>> 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                 C   r   )Nr   Tr   r-   r,   rr1   r3   r%   r%   r&   r5     r   znhypergeom_gen._shape_infoc                 C      d|fS r   r%   )r4   r   r,   r   r%   r%   r&   rD     r   znhypergeom_gen._get_supportc                 C   sD   |dk||k@ |dk@ ||| k@ }|t |t |@ t |@ M }|S r   r>   )r4   r   r,   r   r   r%   r%   r&   r@     s   $znhypergeom_gen._argcheckNc                    s"   t  fdd}||||||dS )Nc                    sl     | ||\}}t||d } || ||}t||ddd}	|	|j|dt}
|d u r4|
 S |
S )Nr   nextextrapolate)kind
fill_valuer   )	supportr!   r   r   r   uniformastypeintitem)r   r,   r   r:   r;   rC   r   ksr   ppfrvsr3   r%   r&   _rvs1  s   z"nhypergeom_gen._rvs.<locals>._rvs1r   _vectorize_rvs_over_shapes)r4   r   r,   r   r:   r;   r  r%   r3   r&   r<     s   znhypergeom_gen._rvsc                 C   s2   |dk|dk@ }t | ||||fdd dd}|S )Nr   c                 S   sv   t | d | t | | d t ||  d || | d  t || |  d d t |d || d  t |d d S rE   r   )rI   r   r,   r   r%   r%   r&   <lambda>  s   z(nhypergeom_gen._logpmf.<locals>.<lambda>        )	fillvalue)r	   )r4   rI   r   r,   r   r   r   r%   r%   r&   rK     s   znhypergeom_gen._logpmfc                 C   r   r    r   )r4   rI   r   r,   r   r%   r%   r&   rS     s   znhypergeom_gen._pmfc                 C   s   d| d| d| }}}|| || d  }||d  | || d || d   d||| d    }d\}}||||fS )Nr   r   r   rb   r%   )r4   r   r,   r   ri   rj   rk   rl   r%   r%   r&   rm     s
   <znhypergeom_gen._statsrb   )rv   rw   rx   ry   r5   rD   r@   r<   rK   rS   rm   r%   r%   r%   r&   r   _  s    c

r   
nhypergeomc                   @   :   e Zd ZdZdd ZdddZdd Zd	d
 Zd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                 C   r~   r   r   r3   r%   r%   r&   r5     r   zlogser_gen._shape_infoNc                 C   r   r   )	logseriesr   r%   r%   r&   r<     r   zlogser_gen._rvsc                 C   s   |dk|dk @ S r=   r%   r   r%   r%   r&   r@   "  r   zlogser_gen._argcheckc                 C   s"   t || d | t|  S Nr   )r!   r   r   r   r   r%   r%   r&   rS   %     "zlogser_gen._pmfc                 C   s  t | }||d  | }| | |d d  }|||  }| | d|  d| d  }|d| |  d|d   }|t|d }| | d|d d  d| |d d   d| | |d d    }	|	d| |  d| | |  d|d   }
|
|d  d }||||fS )	Nr   r   r         ?r   r      r   )r   r   r!   r   )r4   r.   r   ri   mu2prj   mu3pmu3rk   mu4pmu4rl   r%   r%   r&   rm   )  s   :,zlogser_gen._statsrb   )	rv   rw   rx   ry   r5   r<   r@   rS   rm   r%   r%   r%   r&   r	    s    
r	  logserzA logarithmicc                   @   sZ   e Zd ZdZdd Zdd ZdddZd	d
 Zdd Zdd Z	dd Z
dd Zd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                 C      t dddtjfdgS )Nri   Fr   r-   r1   r3   r%   r%   r&   r5   U  r   zpoisson_gen._shape_infoc                 C   s   |dkS r   r%   )r4   ri   r%   r%   r&   r@   Y  r   zpoisson_gen._argcheckNc                 C   s   | ||S r    poisson)r4   ri   r:   r;   r%   r%   r&   r<   \  r   zpoisson_gen._rvsc                 C   s    t ||t|d  | }|S rE   )r   rG   rF   )r4   rI   ri   Pkr%   r%   r&   rK   _  s   zpoisson_gen._logpmfc                 C      t | ||S r    r   )r4   rI   ri   r%   r%   r&   rS   c  s   zpoisson_gen._pmfc                 C      t |}t||S r    )r   r   pdtrr4   r$   ri   rI   r%   r%   r&   rW   g     zpoisson_gen._cdfc                 C   r  r    )r   r   pdtrcr  r%   r%   r&   rZ   k  r  zpoisson_gen._sfc                 C   s>   t t||}t|d d}t||}t||k||S r   )r   r   pdtrikr!   r   r  r   )r4   r_   ri   rr   vals1r   r%   r%   r&   r`   o  s   zpoisson_gen._ppfc                 C   sN   |}t |}|dk}t||fdd t j}t||fdd t j}||||fS )Nr   c                 S   s   t d|  S r  r   r#   r%   r%   r&   r  y  s    z$poisson_gen._stats.<locals>.<lambda>c                 S   s   d|  S r  r%   r#   r%   r%   r&   r  z  s    )r!   r   r	   r2   )r4   ri   rj   tmp
mu_nonzerork   rl   r%   r%   r&   rm   u  s   
zpoisson_gen._statsrb   )rv   rw   rx   ry   r5   r@   r<   rK   rS   rW   rZ   r`   rm   r%   r%   r%   r&   r  <  s    
r  r  z	A Poisson)r{   r   c                   @   sb   e Zd ZdZdd Zdd Zdd Zdd	 Zd
d Zdd Z	dd Z
dddZdd Zd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                 C   r  )NlambdaFr   r   r1   r3   r%   r%   r&   r5     r   zplanck_gen._shape_infoc                 C      |dkS r   r%   )r4   lambda_r%   r%   r&   r@     r   zplanck_gen._argcheckc                 C   s   t |  t| |  S r    )r   r   )r4   rI   r'  r%   r%   r&   rS     r   zplanck_gen._pmfc                 C   s   t |}t| |d   S rE   )r   r   r4   r$   r'  rI   r%   r%   r&   rW     r   zplanck_gen._cdfc                 C   r  r    )r   r   )r4   r$   r'  r%   r%   r&   rZ     r   zplanck_gen._sfc                 C   s   t |}| |d  S rE   r   r(  r%   r%   r&   r     rX   zplanck_gen._logsfc                 C   sL   t d| t|  d }|d j| | }| ||}t||k||S )N      r   )r   r   cliprD   rW   r!   r   )r4   r_   r'  rr   r!  r   r%   r%   r&   r`     s   zplanck_gen._ppfNc                 C   s   t |  }|j||dd S )Nr   r   )r   r   )r4   r'  r:   r;   r.   r%   r%   r&   r<     s   zplanck_gen._rvsc                 C   sP   dt | }t| t | d  }dt|d  }ddt|  }||||fS )Nr   r   r   r  )r   r   r   )r4   r'  ri   rj   rk   rl   r%   r%   r&   rm     s
   zplanck_gen._statsc                 C   s&   t |  }|t|  | t| S r    )r   r   r   )r4   r'  Cr%   r%   r&   rs     s   zplanck_gen._entropyrb   )rv   rw   rx   ry   r5   r@   rS   rW   rZ   r   r`   r<   rm   rs   r%   r%   r%   r&   r$    s    
r$  planckzA discrete exponential c                   @   H   e Zd ZdZdd Zdd Zdd Zdd	 Zd
d Zdd Z	d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                 C   s(   t dddtjfdt dddtjfdgS )Nr'  Fr   r   r   Tr1   r3   r%   r%   r&   r5        zboltzmann_gen._shape_infoc                 C   s   |dk|dk@ t |@ S r   r>   r4   r'  r   r%   r%   r&   r@     r   zboltzmann_gen._argcheckc                 C   s   | j |d fS rE   rB   r1  r%   r%   r&   rD     r(   zboltzmann_gen._get_supportc                 C   s2   dt |  dt | |   }|t | |  S rE   r   )r4   rI   r'  r   factr%   r%   r&   rS     s    zboltzmann_gen._pmfc                 C   s0   t |}dt| |d   dt| |   S rE   )r   r   )r4   r$   r'  r   rI   r%   r%   r&   rW     s   (zboltzmann_gen._cdfc                 C   sd   |dt | |   }td| td|  d }|d dtj}| |||}t||k||S )Nr   r*  r  )r   r   r   r+  r!   r2   rW   r   )r4   r_   r'  r   qnewrr   r!  r   r%   r%   r&   r`     s
   zboltzmann_gen._ppfc                 C   s  t | }t | | }|d|  || d|   }|d| d  || | d| d   }d| d|  }||d  || |  }|d|  |d  |d | d|   }	|	|d  }	|dd|  ||   |d  |d | dd|  ||    }
|
| | }
|||	|
fS )Nr   r   r   r   r  r  r2  )r4   r'  r   zzNri   rj   trmtrm2rk   rl   r%   r%   r&   rm     s   
((@zboltzmann_gen._statsN)rv   rw   rx   ry   r5   r@   rD   rS   rW   r`   rm   r%   r%   r%   r&   r/    s    r/  	boltzmannz!A truncated discrete exponential )r{   rC   r   c                   @   sZ   e Zd ZdZdd Zdd Zdd Zdd	 Zd
d Zdd Z	dd Z
dddZd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                 C   s0   t ddtj tjfdt ddtj tjfdgS )NlowTr   highr1   r3   r%   r%   r&   r5   &  s   zrandint_gen._shape_infoc                 C   s   ||kt |@ t |@ S r    r>   r4   r;  r<  r%   r%   r&   r@   *  r   zrandint_gen._argcheckc                 C   s   ||d fS rE   r%   r=  r%   r%   r&   rD   -  r   zrandint_gen._get_supportc                 C   s,   t |||  }t ||k||k @ |dS )Nr  )r!   	ones_liker   )r4   rI   r;  r<  r.   r%   r%   r&   rS   0  s   zrandint_gen._pmfc                 C   s   t |}|| d ||  S r  r)  )r4   r$   r;  r<  rI   r%   r%   r&   rW   5  r   zrandint_gen._cdfc                 C   sH   t |||  | d }|d ||}| |||}t||k||S rE   )r   r+  rW   r!   r   )r4   r_   r;  r<  rr   r!  r   r%   r%   r&   r`   9  s   zrandint_gen._ppfc           
      C   sj   t |t |}}|| d d }|| }|| d d }d}d|| d  || d  }	||||	fS )Nr   r   r   g      (@r  g333333)r!   r   )
r4   r;  r<  m2m1ri   drj   rk   rl   r%   r%   r&   rm   ?  s   zrandint_gen._statsNc                 C   sr   t |jdkrt |jdkrt||||dS |dur(t ||}t ||}t jtt|t jgd}|||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;   randintr%   r%   r&   r<   H  s    
zrandint_gen._rvsc                 C   s   t || S r    )r   r=  r%   r%   r&   rs   Y  r   zrandint_gen._entropyrb   )rv   rw   rx   ry   r5   r@   rD   rS   rW   r`   rm   r<   rs   r%   r%   r%   r&   r:    s    
	r:  rF  z#A discrete uniform (random integer)c                   @   r  )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`.

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

    c                 C   r  )NrC   Fr   r   r1   r3   r%   r%   r&   r5     r   zzipf_gen._shape_infoNc                 C   r   r   )zipf)r4   rC   r:   r;   r%   r%   r&   r<     r(   zzipf_gen._rvsc                 C   r&  rE   r%   r4   rC   r%   r%   r&   r@     r   zzipf_gen._argcheckc                 C   s   dt |d ||  }|S Nr   r   r   r   )r4   rI   rC   r  r%   r%   r&   rS     s   zzipf_gen._pmfc                 C   s    t ||d k||fdd tjS )Nr   c                 S   s   t | | dt | d S rE   rK  )rC   r,   r%   r%   r&   r        z zipf_gen._munp.<locals>.<lambda>)r	   r!   r2   )r4   r,   rC   r%   r%   r&   _munp  s
   zzipf_gen._munprb   )	rv   rw   rx   ry   r5   r<   r@   rS   rM  r%   r%   r%   r&   rG  b  s    *
rG  rH  zA Zipfc                 C   s   t |dt || d  S )z"Generalized harmonic number, a > 1r   )r   r,   rC   r%   r%   r&   _gen_harmonic_gt1  s   rO  c                 C   sf   t | s| S t | }t j|td}t j|ddtdD ]}|| k}||  d|||   7  < q|S )z#Generalized harmonic number, a <= 1dtyper   r   )r!   r:   max
zeros_likefloatr   )r,   rC   n_maxoutimaskr%   r%   r&   _gen_harmonic_leq1  s   

rZ  c                 C   s(   t | |\} }t|dk| |fttdS )zGeneralized harmonic numberr   ff2)r!   r   r	   rO  rZ  rN  r%   r%   r&   _gen_harmonic  s   r^  c                   @   r.  )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`.

    >>> 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                 C   s(   t dddtjfdt dddtjfdgS )NrC   Fr   r-   r,   Tr   r1   r3   r%   r%   r&   r5     r0  zzipfian_gen._shape_infoc                 C   s"   |dk|dk@ |t j|tdk@ S )Nr   rP  )r!   r   r   r4   rC   r,   r%   r%   r&   r@     r  zzipfian_gen._argcheckc                 C   r   rE   r%   r`  r%   r%   r&   rD     r   zzipfian_gen._get_supportc                 C   s   dt || ||  S r  r^  r4   rI   rC   r,   r%   r%   r&   rS     r   zzipfian_gen._pmfc                 C   s   t ||t || S r    ra  rb  r%   r%   r&   rW     r   zzipfian_gen._cdfc                 C   s:   |d }|| t ||t ||  d || t ||  S rE   ra  rb  r%   r%   r&   rZ     s   zzipfian_gen._sfc                 C   s   t ||}t ||d }t ||d }t ||d }t ||d }|| }|| |d  }	|d }
|	|
 }|| d| | |d   d|d  |d   |d  }|d | d|d  | |  d| |d  |  d|d   |	d  }|d8 }||||fS )Nr   r   r   r  r  r   ra  )r4   rC   r,   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2rk   rl   r%   r%   r&   rm     s"   
82
zzipfian_gen._statsN)rv   rw   rx   ry   r5   r@   rD   rS   rW   rZ   rm   r%   r%   r%   r&   r_    s    -r_  zipfianz	A Zipfianc                   @   sJ   e Zd ZdZdd Zdd Zdd Zdd	 Zd
d Zdd Z	d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                 C   r  )NrC   Fr   r   r1   r3   r%   r%   r&   r5   0  r   zdlaplace_gen._shape_infoc                 C   s   t |d t| t|  S Nr   )r   r   abs)r4   rI   rC   r%   r%   r&   rS   3  s   zdlaplace_gen._pmfc                 C   s0   t |}dd }dd }t|dk||f||dS )Nc                 S   s   dt | |  t |d   S rJ  r2  rI   rC   r%   r%   r&   r  9  s    z#dlaplace_gen._cdf.<locals>.<lambda>c                 S   s   t || d  t |d  S rE   r2  rp  r%   r%   r&   r  :  rL  r   r[  )r   r	   )r4   r$   rC   rI   r\  r]  r%   r%   r&   rW   7  s   zdlaplace_gen._cdfc                 C   st   dt | }tt|ddt |   k t|| | d td| |  | }|d }t| |||k||S )Nr   r   )r   r   r!   r   r   rW   )r4   r_   rC   constrr   r!  r%   r%   r&   r`   =  s   zdlaplace_gen._ppfc                 C   s\   t |}d| |d d  }d| |d d|  d  |d d  }d|d||d  d fS )Nr   r   r   g      $@r  r  r   r2  )r4   rC   eark  r  r%   r%   r&   rm   E  s   (zdlaplace_gen._statsc                 C   s   |t | tt|d  S rn  )r   r   r   rI  r%   r%   r&   rs   K  s   zdlaplace_gen._entropyNc                 C   s8   t t |  }|j||d}|j||d}|| S r   )r!   r   r   r   )r4   rC   r:   r;   probOfSuccessr$   yr%   r%   r&   r<   N  s   zdlaplace_gen._rvsrb   )rv   rw   rx   ry   r5   rS   rW   r`   rm   rs   r<   r%   r%   r%   r&   rm    s    rm  dlaplacezA discrete Laplacianc                   @   r  )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                 C   s(   t dddtjfdt dddtjfdgS )Nrh  Fr   r   rk  r1   r3   r%   r%   r&   r5     r0  zskellam_gen._shape_infoNc                 C   s   |}| ||| || S r    r  )r4   rh  rk  r:   r;   r,   r%   r%   r&   r<     s   

zskellam_gen._rvsc              	   C   sN   t |dk td| dd|  d| d td| dd|  d| d }|S )Nr   r   r   )r!   r   r   r4   r$   rh  rk  pxr%   r%   r&   rS     s
   
zskellam_gen._pmfc              
   C   sN   t |}t|dk td| d| d| dtd| d|d  d|  }|S )Nr   r   r   )r   r!   r   r   rw  r%   r%   r&   rW     s   
zskellam_gen._cdfc                 C   s4   || }|| }|t |d  }d| }||||fS )Nr   r   r   )r4   rh  rk  meanrj   rk   rl   r%   r%   r&   rm     s
   zskellam_gen._statsrb   )	rv   rw   rx   ry   r5   r<   rS   rW   rm   r%   r%   r%   r&   rv  h  s    
rv  skellamz	A Skellamc                   @   sZ   e Zd ZdZdd ZdddZdd Zd	d
 Zdd Zdd Z	dd Z
dd Zd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                 C   r  )NalphaFr   r   r1   r3   r%   r%   r&   r5     r   zyulesimon_gen._shape_infoNc                 C   s6   | |}| |}t| tt| |   }|S r    )standard_exponentialr   r   r   )r4   r}  r:   r;   E1E2ansr%   r%   r&   r<     s   

zyulesimon_gen._rvsc                 C   s   |t ||d  S rE   r   r   r4   r$   r}  r%   r%   r&   rS     r   zyulesimon_gen._pmfc                 C   r&  r   r%   )r4   r}  r%   r%   r&   r@     r   zyulesimon_gen._argcheckc                 C   s   t |t||d  S rE   r   r   r   r  r%   r%   r&   rK     r   zyulesimon_gen._logpmfc                 C   s   d|t ||d   S rE   r  r  r%   r%   r&   rW     r   zyulesimon_gen._cdfc                 C   s   |t ||d  S rE   r  r  r%   r%   r&   rZ     r   zyulesimon_gen._sfc                 C   s   t |t||d  S rE   r  r  r%   r%   r&   r     r   zyulesimon_gen._logsfc                 C   s  t |dkt j||d  }t |dk|d |d |d d   t j}t |dkt j|}t |dkt|d |d d  ||d   t j}t |dkt j|}t |dk|d |d d|  d ||d  |d    t j}t |dkt j|}||||fS )Nr   r   r   r   r  1      )r!   r   r2   nanr   )r4   r}  ri   rk  rk   rl   r%   r%   r&   rm     s*   

"
zyulesimon_gen._statsrb   )rv   rw   rx   ry   r5   r<   rS   r@   rK   rW   rZ   r   rm   r%   r%   r%   r&   r|    s    !
r|  	yulesimon)r{   rC   c                    s    fdd}|S )z?Decorator that vectorizes _rvs method to work on ndarray shapesc                    s   t |d j| \}}t| } t|}t|}t|r)g || |R  S t| }t|j}t||  || f}t	|||}tj
| |   D ] g  fdd|D ||R  | < qOt	|||S )Nr   c                    s   g | ]	}t |  qS r%   )r!   squeeze).0argrX  r%   r&   
<listcomp>  s    z<_vectorize_rvs_over_shapes.<locals>._rvs.<locals>.<listcomp>)r   shaper!   arrayallemptyr   ndimhstackmoveaxisndindex)r:   r;   args
_rvs1_size_rvs1_indicesrW  j0j1r  r  r&   r<     s"   




z(_vectorize_rvs_over_shapes.<locals>._rvsr%   )r  r<   r%   r  r&   r    s   	r  c                   @   sJ   e Zd ZdZdZdZdd Zdd Zdd Zdd	d
Z	dd Z
dd ZdS )_nchypergeom_genzA noncentral hypergeometric discrete random variable.

    For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.

    Nc                 C   sL   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   r%   r%   r&   r5   )  s
   z_nchypergeom_gen._shape_infoc           	      C   s<   |||}}}|| }t d|| }t ||}||fS r   r   )	r4   r   r,   r   r  r@  r?  x_minx_maxr%   r%   r&   rD   /  s
   z_nchypergeom_gen._get_supportc                 C   s   t |t |}}t |t |}}|t|k|dk@ }|t|k|dk@ }|t|k|dk@ }|dk}||k}	||k}
||@ |@ |@ |	@ |
@ S r   )r!   r   r   r   )r4   r   r,   r   r  cond1cond2cond3cond4cond5cond6r%   r%   r&   r@   6  s   z_nchypergeom_gen._argcheckc                    s$   t  fdd}|||||||dS )Nc           
         s<   t |}t }t| j}|||| |||}	|	|}	|	S r    )r!   prodr   getattrrvs_namereshape)
r   r,   r   r  r:   r;   lengthurnrv_genr   r3   r%   r&   r  C  s   

z$_nchypergeom_gen._rvs.<locals>._rvs1r   r  )r4   r   r,   r   r  r:   r;   r  r%   r3   r&   r<   A  s   z_nchypergeom_gen._rvsc                    sR   t |||||\}}}}}|jdkrt |S t j fdd}||||||S )Nr   c                    s     ||||d}|| S Ng-q=)distprobability)r$   r   r,   r   r  r  r3   r%   r&   _pmf1T  s   
z$_nchypergeom_gen._pmf.<locals>._pmf1)r!   r   r:   
empty_likerD  )r4   r$   r   r,   r   r  r  r%   r3   r&   rS   N  s   

z_nchypergeom_gen._pmfc                    sL   t j fdd}d|v sd|v r|||||nd\}}d\}	}
|||	|
fS )Nc                    s     ||| |d}| S r  )r  rh   )r   r,   r   r  r  r3   r%   r&   	_moments1]  s   z*_nchypergeom_gen._stats.<locals>._moments1r   vrb   )r!   rD  )r4   r   r,   r   r  rh   r  r   r  rc   rI   r%   r3   r&   rm   [  s   z_nchypergeom_gen._statsrb   )rv   rw   rx   ry   r  r  r5   rD   r@   r<   rS   rm   r%   r%   r%   r&   r    s    
r  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)rv   rw   rx   ry   r  r   r  r%   r%   r%   r&   r  h      Ir  nchypergeom_fisherz$A Fisher's noncentral hypergeometricc                   @   r  )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)rv   rw   rx   ry   r  r   r  r%   r%   r%   r&   r    r  r  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)`	functoolsr   r   scipyr   scipy.specialr   r   r   r   rF   r   scipy._lib._utilr	   r
   scipy.interpolater   numpyr   r   r   r   r   r   r   r   r   r   r!   _distn_infrastructurer   r   r   r   r   r   scipy.stats._booststatsrP   
_biasedurnr   r   r   r'   r)   rz   r}   r   r   r   r   r   r   r   r   r   r   r  r	  r  r  r  r$  r-  r/  r9  r:  rF  rG  rH  rO  rZ  r^  r_  rl  rm  r2   ru  rv  r{  r|  r  r  r  r  r  r  r  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r%   r%   r%   r&   <module>   s   0 
RA
R
rE 
 
8BG?O@WK=N&INN