
    _MhG                        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 m!Z!m"Z" dd
l#m$Z$m%Z%m&Z& ddl'm(Z( d dl)mc m*Z+  G d de          Z, e,d          Z- G d de,          Z. e.dd          Z/ G d de          Z0 e0d          Z1 G d de          Z2 e2d          Z3 G d de          Z4 e4d          Z5 G d de          Z6 e6ddd !          Z7 G d" d#e          Z8 e8d$          Z9 G d% d&e          Z: e:d'          Z; G d( d)e          Z< e<dd*d+!          Z= G d, d-e          Z> e>d.d/0          Z? G d1 d2e          Z@ e@d d3d4!          ZA G d5 d6e          ZB eBd7d d89          ZC G d: d;e          ZD eDd<d=0          ZE G d> d?e          ZF eFdd@dA!          ZGdB ZHdC ZIdD ZJ G dE dFe          ZK eKddGdH!          ZL G dI dJe          ZM eMejN         dKdL!          ZO G dM dNe          ZP ePdOdPdQR          ZQdkdSZRdldUZSdmdVZTeQePcZUZVeRW                    eUeV          eQ_R        eSW                    eUeV          eQ_S        eTW                    eUeV          eQ_T         G dW dXe"          ZX G dY dZe          ZY eYejN         d[d\!          ZZ G d] d^e          Z[ e[d_d`          Z\ G da dbe          Z] G dc dde]          Z^ e^dedf0          Z_ G dg dhe]          Z` e`didj0          Za eb ec            d                                e                                          Zf eefe          \  ZgZhegehz   ZidS )n    )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_vectorize_rvs_over_shapes
_ShapeInfo_isintegralrv_discrete_frozen)_PyFishersNCHypergeometric_PyWalleniusNCHypergeometric_PyStochasticLib3)_poisson_binomc                   ^    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_gena2  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.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pmf``, ``cdf``, ``sf``, ``ppf`` and ``isf``
    methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(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   npinfselfs    \/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/stats/_discrete_distns.py_shape_infozbinom_gen._shape_infoA   4    3q"&k=AA3v|<<> 	>    Nc                 0    |                     |||          S N)binomialr/   r&   r(   sizerandom_states        r0   _rvszbinom_gen._rvsE   s    $$Q4000r3   c                 J    |dk    t          |          z  |dk    z  |dk    z  S Nr   r   r   r/   r&   r(   s      r0   	_argcheckzbinom_gen._argcheckH   s)    Q+a..(AF3qAv>>r3   c                     | j         |fS r5   ar>   s      r0   _get_supportzbinom_gen._get_supportK   s    vqyr3   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)r/   xr&   r(   kcombilns         r0   _logpmfzbinom_gen._logpmfN   sl    !HH1::qseAaCEll!:;q!,,,wqsQB/G/GGGr3   c                 .    t          j        |||          S r5   )scu
_binom_pmfr/   rI   r&   r(   s       r0   _pmfzbinom_gen._pmfS   s    ~aA&&&r3   c                 L    t          |          }t          j        |||          S r5   )r   rN   
_binom_cdfr/   rI   r&   r(   rJ   s        r0   _cdfzbinom_gen._cdfW   !    !HH~aA&&&r3   c                 L    t          |          }t          j        |||          S r5   )r   rN   	_binom_sfrT   s        r0   _sfzbinom_gen._sf[   s!    !HH}Q1%%%r3   c                 .    t          j        |||          S r5   )rN   
_binom_isfrP   s       r0   _isfzbinom_gen._isf_       ~aA&&&r3   c                 .    t          j        |||          S r5   )rN   
_binom_ppfr/   qr&   r(   s       r0   _ppfzbinom_gen._ppfb   r]   r3   mvc                 x   ||z  }||t          j        |          z  z
  }d\  }}d|v rO|t          j        |          z
  }t          j        ||z            }	t          j        |	          }
d|z  |	z  }|
|z
  }d|v r:|t          j        |          z
  }||z  }t          j        |          }
d|z  }|
|z
  }||||fS )NNNs       @rJ         @)r,   squarer   
reciprocal)r/   r&   r(   momentsmuvarg1g2pqnpq_sqrtt1t2npqs                r0   _statszbinom_gen._statse   s    U1ry||##B'>>RYq\\!Bwq2vHx((B'X%BbB'>>RYq\\!Bb&Cs##BQBbB3Br3   c                     t           j        d|dz            }|                     |||          }t          j        t	          |          d          S )Nr   r   axis)r,   r_rQ   sumr   )r/   r&   r(   rJ   valss        r0   _entropyzbinom_gen._entropyw   sE    E!AE'NyyAq!!vd4jjq))))r3   re   rc   __name__
__module____qualname____doc__r1   r:   r?   rC   rL   rQ   rU   rY   r\   rb   ru   r|    r3   r0   r#   r#      s        " "F> > >1 1 1 1? ? ?  H H H
' ' '' ' '& & &' ' '' ' '   $* * * * *r3   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)   r*   r   r.   s    r0   r1   zbernoulli_gen._shape_info       3v|<<==r3   Nc                 @    t                               | d|||          S )Nr   r8   r9   )r#   r:   r/   r(   r8   r9   s       r0   r:   zbernoulli_gen._rvs   s    ~~dAqt,~OOOr3   c                     |dk    |dk    z  S r<   r   r/   r(   s     r0   r?   zbernoulli_gen._argcheck   s    Q16""r3   c                     | j         | j        fS r5   )rB   br   s     r0   rC   zbernoulli_gen._get_support   s    vtv~r3   c                 :    t                               |d|          S rE   )r   rL   r/   rI   r(   s      r0   rL   zbernoulli_gen._logpmf   s    }}Q1%%%r3   c                 :    t                               |d|          S rE   )r   rQ   r   s      r0   rQ   zbernoulli_gen._pmf   s     zz!Q"""r3   c                 :    t                               |d|          S rE   )r   rU   r   s      r0   rU   zbernoulli_gen._cdf       zz!Q"""r3   c                 :    t                               |d|          S rE   )r   rY   r   s      r0   rY   zbernoulli_gen._sf   s    yyAq!!!r3   c                 :    t                               |d|          S rE   )r   r\   r   s      r0   r\   zbernoulli_gen._isf   r   r3   c                 :    t                               |d|          S rE   )r   rb   )r/   ra   r(   s      r0   rb   zbernoulli_gen._ppf   r   r3   c                 8    t                               d|          S rE   )r   ru   r   s     r0   ru   zbernoulli_gen._stats   s    ||Aq!!!r3   c                 F    t          |          t          d|z
            z   S rE   )r   r   s     r0   r|   zbernoulli_gen._entropy   s    Awwac""r3   re   r~   r   r3   r0   r   r      s         0> > >P P P P# # #  & & &# # #
# # #" " "# # ## # #" " "# # # # #r3   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'   rB   FFFr   r+   r.   s    r0   r1   zbetabinom_gen._shape_info   S    3q"&k=AA326{NCC326{NCCE 	Er3   Nc                 ^    |                     |||          }|                    |||          S r5   )betar6   r/   r&   rB   r   r8   r9   r(   s          r0   r:   zbetabinom_gen._rvs   s1    aD))$$Q4000r3   c                 
    d|fS Nr   r   r/   r&   rB   r   s       r0   rC   zbetabinom_gen._get_support       !tr3   c                 J    |dk    t          |          z  |dk    z  |dk    z  S r   r=   r   s       r0   r?   zbetabinom_gen._argcheck   )    Q+a..(AE2a!e<<r3   c                     t          |          }t          |dz              t          ||z
  dz   |dz             z
  }|t          ||z   ||z
  |z             z   t          ||          z
  S rE   )r   r   r   r/   rI   r&   rB   r   rJ   rK   s          r0   rL   zbetabinom_gen._logpmf   sg    !HHq1u::+q1uqy!a% 8 88Aq1uqy111F1aLL@@r3   c                 L    t          |                     ||||                    S r5   r   rL   r/   rI   r&   rB   r   s        r0   rQ   zbetabinom_gen._pmf   "    4<<1a++,,,r3   rc   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   re   rf         ?   rJ            )r   astypedtype)r/   r&   rB   r   rk   e_pe_qrl   rm   rn   ro   s              r0   ru   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r3   re   r}   )r   r   r   r   r1   r:   rC   r?   rL   rQ   ru   r   r3   r0   r   r      s        " "FE E E
1 1 1 1  = = =A A A
- - -     r3   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}

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pmf``, ``cdf``, ``sf``, ``ppf``, ``isf``
    and ``stats`` methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(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%   r+   r.   s    r0   r1   znbinom_gen._shape_infoT  r2   r3   Nc                 0    |                     |||          S r5   )negative_binomialr7   s        r0   r:   znbinom_gen._rvsX  s    --aD999r3   c                 *    |dk    |dk    z  |dk    z  S r<   r   r>   s      r0   r?   znbinom_gen._argcheck[  s    A!a% AF++r3   c                 .    t          j        |||          S r5   )rN   _nbinom_pmfrP   s       r0   rQ   znbinom_gen._pmf^  s    q!Q'''r3   c                     t          ||z             t          |dz             z
  t          |          z
  }||t          |          z  z   t          j        ||           z   S rE   )rF   r   r   rH   )r/   rI   r&   r(   coeffs        r0   rL   znbinom_gen._logpmfb  sS    ac

U1Q3ZZ'%((2qQx'/!aR"8"888r3   c                 L    t          |          }t          j        |||          S r5   )r   rN   _nbinom_cdfrT   s        r0   rU   znbinom_gen._cdff  s!    !HHq!Q'''r3   c                 x   t          |          }t          j        |||          \  }}}|                     |||          }|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 rE   )r,   r   r   betainc)rJ   r&   r(   s      r0   f1znbinom_gen._logcdf.<locals>.f1o  s+    8W_QUAq1u===>>>r3   ignore)divide)r   r,   broadcast_arraysrU   errstater   )	r/   rI   r&   r(   rJ   cdfcondr   logcdfs	            r0   _logcdfznbinom_gen._logcdfj  s   !HH%aA..1aii1a  Sy	? 	? 	? [))) 	/ 	/2agqw$88F4LF3u:..FD5M	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ 	/ s   !AB//B36B3c                 L    t          |          }t          j        |||          S r5   )r   rN   
_nbinom_sfrT   s        r0   rY   znbinom_gen._sfy  rV   r3   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   rN   _nbinom_isfrP   s       r0   r\   znbinom_gen._isf}      [h''' 	, 	,?1a++	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	, 	,   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   rN   _nbinom_ppfr`   s       r0   rb   znbinom_gen._ppf  r   r   c                     t          j        ||          t          j        ||          t          j        ||          t          j        ||          fS r5   )rN   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessr>   s      r0   ru   znbinom_gen._stats  sL    Q"" A&& A&&'1--	
 	
r3   re   )r   r   r   r   r1   r:   r?   rQ   rL   rU   r   rY   r\   rb   ru   r   r3   r0   r   r     s        9 9t> > >: : : :, , ,( ( (9 9 9( ( (  ' ' ', , ,, , ,
 
 
 
 
r3   r   nbinomc                   :    e Zd ZdZd Zd
dZd Zd Zd Zdd	Z	dS )betanbinom_genaK  A beta-negative-binomial discrete random variable.

    %(before_notes)s

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

    The probability mass function for `betanbinom` is:

    .. math::

       f(k) = \binom{n + k - 1}{k} \frac{B(a + n, b + k)}{B(a, b)}

    for :math:`k \ge 0`, :math:`n \geq 0`, :math:`a > 0`,
    :math:`b > 0`, where :math:`B(a, b)` is the beta function.

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

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

    %(after_notes)s

    .. versionadded:: 1.12.0

    See Also
    --------
    betabinom : Beta binomial distribution

    %(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 r   r+   r.   s    r0   r1   zbetanbinom_gen._shape_info  r   r3   Nc                 ^    |                     |||          }|                    |||          S r5   )r   r   r   s          r0   r:   zbetanbinom_gen._rvs  s1    aD))--aD999r3   c                 J    |dk    t          |          z  |dk    z  |dk    z  S r   r=   r   s       r0   r?   zbetanbinom_gen._argcheck  r   r3   c                     t          |          }t          j        ||z              t          ||dz             z
  }|t          ||z   ||z             z   t          ||          z
  S rE   )r   r,   r   r   r   s          r0   rL   zbetanbinom_gen._logpmf  s]    !HH6!a%==.6!QU#3#33Aq1u---q!<<r3   c                 L    t          |                     ||||                    S r5   r   r   s        r0   rQ   zbetanbinom_gen._pmf  r   r3   rc   c                 `   d }t          |dk    |||f|t          j                  }d }t          |dk    |||f|t          j                  }d\  }}	d }
d|v r$t          |d	k    |||f|
t          j                  }d
 }d|v r$t          |dk    |||f|t          j                  }	||||	fS )Nc                     | |z  |dz
  z  S Nr   r   r&   rB   r   s      r0   meanz#betanbinom_gen._stats.<locals>.mean  s    q5AF##r3   r   )f	fillvaluec                 N    | |z  | |z   dz
  z  ||z   dz
  z  |dz
  |dz
  dz  z  z  S )Nr   rg   r   r   s      r0   rm   z"betanbinom_gen._stats.<locals>.var  sA    EQURZ(AEBJ7B1r6B,.0 1r3   r   re   c                     d| z  |z   dz
  d|z  |z   dz
  z  |dz
  z  t          | |z  | |z   dz
  z  ||z   dz
  z  |dz
  z            z  S )Nr   r         @rg   r   r   s      r0   skewz#betanbinom_gen._stats.<locals>.skew  sq    UQY^A	B72v!%a!eq1urz&:a!ebj&I2v' "  "   !r3   rf   r   c                 z   |dz
  }|dz
  dz  |dz  |d|z  dz
  z  z   d|dz
  z  |z  z   z  d| dz  z  |dz   |dz  z  |dz   |dz
  z  |z  z   d|dz
  dz  z  z   z  z   d|dz
  z  | z  |dz   |dz  z  |dz   |dz
  z  |z  z   d|dz
  dz  z  z   z  z   }|d	z
  |dz
  z  |z  | z  ||z   dz
  z  || z   dz
  z  }||z  |z  dz
  S )
Nrg   r   r   rh   r         @r   r   g      @r   )r&   rB   r   termterm_2denominators         r0   kurtosisz'betanbinom_gen._stats.<locals>.kurtosis  sN   FD2vlaea1q52:.>&>a"f)'* +QU
q2vB&6!b&R:!#$:% '%')QVaK'7'8 99 QVq(b&ArE)QVB,?!,CCa"fr\)*+	+F Fq2v.2Q6!ebj*-.URZ9K &=;.33r3   rJ      r
   r,   r-   )r/   r&   rB   r   rk   r   rl   rm   rn   ro   r   r   s               r0   ru   zbetanbinom_gen._stats  s    	$ 	$ 	$A1ayDBFCCC	1 	1 	1 QAq	SBFCCCB	! 	! 	! '>>AEAq!9GGGB	4 	4 	4 '>>AEAq!9BFKKKB3Br3   re   r}   )
r   r   r   r   r1   r:   r?   rL   rQ   ru   r   r3   r0   r   r     s        # #HE E E
: : : := = == = =
- - -! ! ! ! ! !r3   r   
betanbinomc                   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_gena5  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.

    Note that when drawing random samples, the probability of observations that exceed
    ``np.iinfo(np.int64).max`` increases rapidly as $p$ decreases below $10^{-17}$. For
    $p < 10^{-20}$, almost all observations would exceed the maximum ``int64``; however,
    the output dtype is always ``int64``, so these values are clipped to the maximum.

    %(after_notes)s

    See Also
    --------
    planck

    %(example)s

    c                 (    t          dddd          gS r   r   r.   s    r0   r1   zgeom_gen._shape_info  r   r3   Nc                     |                     ||          }t          j        |j                  j        }t          j        |dk     ||          S )Nr8   r   )	geometricr,   iinfor   maxwhere)r/   r(   r8   r9   resmax_ints         r0   r:   zgeom_gen._rvs  sH    $$QT$22 (39%%)xa#...r3   c                     |dk    |dk    z  S Nr   r   r   r   s     r0   r?   zgeom_gen._argcheck  s    Q1q5!!r3   c                 >    t          j        d|z
  |dz
            |z  S rE   )r,   powerr/   rJ   r(   s      r0   rQ   zgeom_gen._pmf   s!    x!QqS!!A%%r3   c                 T    t          j        |dz
  |           t          |          z   S rE   )r   rH   r   r  s      r0   rL   zgeom_gen._logpmf#  s%    q1uqb))CFF22r3   c                 b    t          |          }t          t          |           |z             S r5   )r   r   r   r/   rI   r(   rJ   s       r0   rU   zgeom_gen._cdf&  s*    !HHeQBiik""""r3   c                 R    t          j        |                     ||                    S r5   )r,   r   _logsfr   s      r0   rY   zgeom_gen._sf*  s     vdkk!Q''(((r3   c                 F    t          |          }|t          |           z  S r5   )r   r   r  s       r0   r  zgeom_gen._logsf-  s    !HHr{r3   c                     t          t          |           t          |           z            }|                     |dz
  |          }t          j        ||k    |dk    z  |dz
  |          S r  )r   r   rU   r,   r  )r/   ra   r(   r{   temps        r0   rb   zgeom_gen._ppf1  s`    E1"IIqb		)**yya##xtax0$q&$???r3   c                     d|z  }d|z
  }||z  |z  }d|z
  t          |          z  }t          j        g d|          d|z
  z  }||||fS )Nr   rg   )r   ir   )r   r,   polyval)r/   r(   rl   qrrm   rn   ro   s          r0   ru   zgeom_gen._stats6  sa    UU1fqj!etBxxZ


A&&A.3Br3   c                 j    t          j        |           t          j        |           d|z
  z  |z  z
  S r   )r,   r   r   r   s     r0   r|   zgeom_gen._entropy>  s/    q		zBHaRLLCE2Q666r3   re   )r   r   r   r   r1   r:   r?   rQ   rL   rU   rY   r  rb   ru   r|   r   r3   r0   r  r    s         B> > >/ / / /" " "& & &3 3 3# # #) ) )  @ @ @
  7 7 7 7 7r3   r  geomzA geometric)rB   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)!}.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pmf``, ``cdf``, ``sf`` and ``stats`` methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    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&   Nr+   r.   s    r0   r1   zhypergeom_gen._shape_info  S    3q"&k=AA3q"&k=AA3q"&k=AAC 	Cr3   Nc                 :    |                     |||z
  ||          S Nr  )hypergeometric)r/   r%  r&   r&  r8   r9   s         r0   r:   zhypergeom_gen._rvs  s#    **1ac14*@@@r3   c                 b    t          j        |||z
  z
  d          t          j        ||          fS r   r,   maximumminimum)r/   r%  r&   r&  s       r0   rC   zhypergeom_gen._get_support  s-    z!QqS'1%%rz!Q'7'777r3   c                     |dk    |dk    z  |dk    z  }|||k    ||k    z  z  }|t          |          t          |          z  t          |          z  z  }|S r   r=   )r/   r%  r&   r&  r   s        r0   r?   zhypergeom_gen._argcheck  s_    A!q&!Q!V,aAF##AQ/+a..@@r3   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 rE   r   )	r/   rJ   r%  r&   r&  totgoodbadresults	            r0   rL   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""# r3   c                 0    t          j        ||||          S r5   )rN   _hypergeom_pmfr/   rJ   r%  r&   r&  s        r0   rQ   zhypergeom_gen._pmf      !!Q1---r3   c                 0    t          j        ||||          S r5   )rN   _hypergeom_cdfr8  s        r0   rU   zhypergeom_gen._cdf  r9  r3   c                 x   d|z  d|z  d|z  }}}||z
  }||dz   z  d|z  ||z
  z  z
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  }||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   rh   r   r   rg   r   )rN   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r/   r%  r&   r&  mro   s         r0   ru   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((#Aq!,,#Aq!,,	
 	
r3   c                     t           j        |||z
  z
  t          ||          dz            }|                     ||||          }t          j        t          |          d          S )Nr   r   rw   )r,   ry   minpmfrz   r   )r/   r%  r&   r&  rJ   r{   s         r0   r|   zhypergeom_gen._entropy  sY    E!q1u+c!Qii!m+,xx1a##vd4jjq))))r3   c                 0    t          j        ||||          S r5   )rN   _hypergeom_sfr8  s        r0   rY   zhypergeom_gen._sf  s     Aq!,,,r3   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,   r   appendr   r   r   aranger   rL   asarrayr/   rJ   r%  r&   r&  r  quantr2  r3  drawk2s              r0   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#r3   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   )rG  r,   r   rH  r   r   logsfrI  r   rL   rJ  rK  s              r0   r   zhypergeom_gen._logcdf  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#r3   re   )r   r   r   r   r1   r:   rC   r?   rL   rQ   rU   ru   r|   rY   r  r   r   r3   r0   r#  r#  E  s        H HRC C C
A A A A8 8 8    . . .. . .
 
 
"* * *
- - -
 
 

 
 
 
 
r3   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&   rr+   r.   s    r0   r1   znhypergeom_gen._shape_infoH  r'  r3   c                 
    d|fS r   r   )r/   r%  r&   rU  s       r0   rC   znhypergeom_gen._get_supportM  r   r3   c                     |dk    ||k    z  |dk    z  |||z
  k    z  }|t          |          t          |          z  t          |          z  z  }|S r   r=   )r/   r%  r&   rU  r   s        r0   r?   znhypergeom_gen._argcheckP  sU    Q16"a1f-ac:AQ/+a..@@r3   Nc                 H     t            fd            } ||||||          S )Nc                 \                        | ||          \  }}t          j        ||dz             }                    || ||          }t	          ||dd          }	 |	|                    |                                        t                    }
||
                                S |
S )Nr   nextextrapolate)kind
fill_valuer  )	supportr,   rI  r   r   uniformr   intitem)r%  r&   rU  r8   r9   rB   r   ksr   ppfrvsr/   s              r0   _rvs1z"nhypergeom_gen._rvs.<locals>._rvs1W  s     <<1a((DAq1ac""B((2q!Q''C3MJJJC#l***5566==cBBC|xxzz!Jr3   r   r   )r/   r%  r&   rU  r8   r9   re  s   `      r0   r:   znhypergeom_gen._rvsU  sD    	#		 		 		 		 
$	#		 uQ14lCCCCr3   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
  |z
  dz             z
  t          ||z
  | z
  dz   d          z   t          |dz   ||z
  dz             z   t          |dz   d          z
  S rE   r1  )rJ   r%  r&   rU  s       r0   <lambda>z(nhypergeom_gen._logpmf.<locals>.<lambda>h  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 r3           )r   )r
   )r/   rJ   r%  r&   rU  r   r5  s          r0   rL   znhypergeom_gen._logpmfe  sN    aAF#TEAq!Q<F F '*+ + + r3   c                 L    t          |                     ||||                    S r5   r   )r/   rJ   r%  r&   rU  s        r0   rQ   znhypergeom_gen._pmfo  s$     4<<1a++,,,r3   c                     d|z  d|z  d|z  }}}||z  ||z
  dz   z  }||dz   z  |z  ||z
  dz   ||z
  dz   z  z  d|||z
  dz   z  z
  z  }d\  }}||||fS )Nr   r   r   re   r   )r/   r%  r&   rU  rl   rm   rn   ro   s           r0   ru   znhypergeom_gen._statst  s     Q$1bda1qSAaCE]1gaiAaCEAaCE?+q1!A;? B3Br3   re   )r   r   r   r   r1   rC   r?   r:   rL   rQ   ru   r   r3   r0   rS  rS    s        b bHC C C
    
D D D D   - - -
    r3   rS  
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   r.   s    r0   r1   zlogser_gen._shape_info  r   r3   Nc                 0    |                     ||          S r)  )	logseriesr   s       r0   r:   zlogser_gen._rvs  s     %%ad%333r3   c                     |dk    |dk     z  S r<   r   r   s     r0   r?   zlogser_gen._argcheck  s    A!a%  r3   c                 f    t          j        ||           dz  |z  t          j        |           z  S r   )r,   r  r   r   r  s      r0   rQ   zlogser_gen._pmf  s/    A$q(7=!+<+<<<r3   c                    t          j        |           }||dz
  z  |z  }| |z  |dz
  dz  z  }|||z  z
  }| |z  d|z   z  d|z
  dz  z  }|d|z  |z  z
  d|dz  z  z   }|t          j        |d          z  }| |z  d|dz
  dz  z  d|z  |dz
  dz  z  z
  d|z  |z  |dz
  dz  z  z   z  }	|	d|z  |z  z
  d|z  |z  |z  z   d|dz  z  z
  }
|
|dz  z  dz
  }||||fS )	Nr   r   r         ?r   r   r  r   )r   r   r,   r  )r/   r(   rU  rl   mu2prm   mu3pmu3rn   mu4pmu4ro   s               r0   ru   zlogser_gen._stats  s@   M1"!c']QrAvS1$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r3   re   )	r   r   r   r   r1   r:   r?   rQ   ru   r   r3   r0   ro  ro    sn         0> > >4 4 4 4
! ! != = =    r3   ro  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 )Nrl   Fr   r'   r+   r.   s    r0   r1   zpoisson_gen._shape_info  s    4BF]CCDDr3   c                     |dk    S r   r   )r/   rl   s     r0   r?   zpoisson_gen._argcheck  s    Qwr3   Nc                 .    |                     ||          S r5   poisson)r/   rl   r8   r9   s       r0   r:   zpoisson_gen._rvs  s    ##B---r3   c                 \    t          j        ||          t          |dz             z
  |z
  }|S rE   )r   rG   rF   )r/   rJ   rl   Pks       r0   rL   zpoisson_gen._logpmf  s,    ]1b!!E!a%LL025	r3   c                 H    t          |                     ||                    S r5   r   )r/   rJ   rl   s      r0   rQ   zpoisson_gen._pmf  s    4<<2&&'''r3   c                 J    t          |          }t          j        ||          S r5   )r   r   pdtrr/   rI   rl   rJ   s       r0   rU   zpoisson_gen._cdf  s    !HH|Ar"""r3   c                 J    t          |          }t          j        ||          S r5   )r   r   pdtrcr  s       r0   rY   zpoisson_gen._sf  s    !HH}Q###r3   c                     t          t          j        ||                    }t          j        |dz
  d          }t          j        ||          }t          j        ||k    ||          S r  )r   r   pdtrikr,   r-  r  r  )r/   ra   rl   r{   vals1r  s         r0   rb   zpoisson_gen._ppf  sY    GN1b))**
4!8Q''|E2&&x	5$///r3   c                     |}t          j        |          }|dk    }t          ||fd t           j                  }t          ||fd t           j                  }||||fS )Nr   c                 &    t          d| z            S r   r   rI   s    r0   ri  z$poisson_gen._stats.<locals>.<lambda>  s    d3q5kk r3   c                     d| z  S r   r   r  s    r0   ri  z$poisson_gen._stats.<locals>.<lambda>  s
    c!e r3   )r,   rJ  r
   r-   )r/   rl   rm   tmp
mu_nonzerorn   ro   s          r0   ru   zpoisson_gen._stats  s_    jnn1W

SF,A,A26JJ
SFOORVDD3Br3   re   )r   r   r   r   r1   r?   r:   rL   rQ   rU   rY   rb   ru   r   r3   r0   r~  r~    s         0E E E  . . . .  ( ( (# # #$ $ $0 0 0    r3   r~  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   r+   r.   s    r0   r1   zplanck_gen._shape_info"  s    8UQKHHIIr3   c                     |dk    S r   r   )r/   lambda_s     r0   r?   zplanck_gen._argcheck%  s    {r3   c                 L    t          |            t          | |z            z  S r5   )r   r   )r/   rJ   r  s      r0   rQ   zplanck_gen._pmf(  s$    whWHQJ//r3   c                 N    t          |          }t          | |dz   z             S rE   )r   r   r/   rI   r  rJ   s       r0   rU   zplanck_gen._cdf+  s(    !HHwh!n%%%%r3   c                 H    t          |                     ||                    S r5   )r   r  )r/   rI   r  s      r0   rY   zplanck_gen._sf/  s    4;;q'**+++r3   c                 2    t          |          }| |dz   z  S rE   r   r  s       r0   r  zplanck_gen._logsf2  s    !HHx1~r3   c                     t          d|z  t          |           z  dz
            } |dz
  j        |                     |           }|                     ||          }t          j        ||k    ||          S )N      r   )r   r   cliprC   rU   r,   r  )r/   ra   r  r{   r  r  s         r0   rb   zplanck_gen._ppf6  sp    DL5!99,Q.//a 1 1' : :<yy((x	5$///r3   Nc                 X    t          |            }|                    ||          dz
  S )Nr  r   )r   r	  )r/   r  r8   r9   r(   s        r0   r:   zplanck_gen._rvs<  s0    G8__%%ad%33c99r3   c                     dt          |          z  }t          |           t          |           dz  z  }dt          |dz            z  }ddt          |          z  z   }||||fS )Nr   r   rg   r  )r   r   r   )r/   r  rl   rm   rn   ro   s         r0   ru   zplanck_gen._statsA  si    uW~~7(mmUG8__q00tGCK   qg3Br3   c                 p    t          |            }|t          |           z  |z  t          |          z
  S r5   )r   r   r   )r/   r  Cs      r0   r|   zplanck_gen._entropyH  s5    G8__sG8}}$Q&Q//r3   re   )r   r   r   r   r1   r?   rQ   rU   rY   r  rb   r:   ru   r|   r   r3   r0   r  r    s         6J J J  0 0 0& & &, , ,  0 0 0: : : :
  0 0 0 0 0r3   r  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 )Nr  Fr   r   r&  Tr+   r.   s    r0   r1   zboltzmann_gen._shape_infof  s<    9ea[.II3q"&k>BBD 	Dr3   c                 <    |dk    |dk    z  t          |          z  S r   r=   r/   r  r&  s      r0   r?   zboltzmann_gen._argcheckj  s     !A&Q77r3   c                     | j         |dz
  fS rE   rA   r  s      r0   rC   zboltzmann_gen._get_supportm  s    vq1u}r3   c                     dt          |           z
  dt          | |z            z
  z  }|t          | |z            z  S rE   r   )r/   rJ   r  r&  facts        r0   rQ   zboltzmann_gen._pmfp  sB     #wh--!C
OO"34C
OO##r3   c                     t          |          }dt          | |dz   z            z
  dt          | |z            z
  z  S rE   )r   r   )r/   rI   r  r&  rJ   s        r0   rU   zboltzmann_gen._cdfv  s@    !HH#wh!n%%%#whqj//(9::r3   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  rj  )r   r   r   r  r,   r-   rU   r  )r/   ra   r  r&  qnewr{   r  r  s           r0   rb   zboltzmann_gen._ppfz  s    !C
OO#$DL3qv;;.q011ac26**yy++x	5$///r3   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   rv  r  r  )r/   r  r&  zzNrl   rm   trmtrm2rn   ro   s              r0   ru   zboltzmann_gen._stats  s,   MM'!__AYqtQrT{"Q
lQqSVQrTAI--tacl#q&1Q3r6!!WS!V^ad2gqtn,$+!A#ac	]36!AqD2Iq2vbe|$<<$Y3Br3   N)r   r   r   r   r1   r?   rC   rQ   rU   rb   ru   r   r3   r0   r  r  P  s         *D D D8 8 8  $ $ $; ; ;0 0 0    r3   r  	boltzmannz!A truncated discrete exponential )r   rB   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

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

    Calculate the first four moments:

    >>> low, high = 7, 31
    >>> mean, var, skew, kurt = randint.stats(low, high, moments='mvsk')

    Display the probability mass function (``pmf``):

    >>> x = np.arange(low - 5, high + 5)
    >>> ax.plot(x, randint.pmf(x, low, high), 'bo', ms=8, label='randint pmf')
    >>> ax.vlines(x, 0, randint.pmf(x, low, high), colors='b', lw=5, alpha=0.5)

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

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

    >>> rv = randint(low, high)
    >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-',
    ...           lw=1, label='frozen pmf')
    >>> ax.legend(loc='lower center')
    >>> plt.show()

    Check the relationship between the cumulative distribution function
    (``cdf``) and its inverse, the percent point function (``ppf``):

    >>> q = np.arange(low, high)
    >>> p = randint.cdf(q, low, high)
    >>> np.allclose(q, randint.ppf(p, low, high))
    True

    Generate random numbers:

    >>> r = randint.rvs(low, high, size=1000)

    c                     t          ddt          j         t          j        fd          t          ddt          j         t          j        fd          gS )NlowTr   highr+   r.   s    r0   r1   zrandint_gen._shape_info  sF    5$"&"&(9>JJ6426'26):NKKM 	Mr3   c                 N    ||k    t          |          z  t          |          z  S r5   r=   r/   r  r  s      r0   r?   zrandint_gen._argcheck  s&    s
k#...T1B1BBBr3   c                     ||dz
  fS rE   r   r  s      r0   rC   zrandint_gen._get_support  s    DF{r3   c                     t          j        |          t          j        |t           j                  |z
  z  }t          j        ||k    ||k     z  |d          S )Nr   rj  )r,   	ones_likerJ  int64r  )r/   rJ   r  r  r(   s        r0   rQ   zrandint_gen._pmf  sK    LOOrz$bh???#EFxca$h/B777r3   c                 <    t          |          }||z
  dz   ||z
  z  S r   r  )r/   rI   r  r  rJ   s        r0   rU   zrandint_gen._cdf  s$    !HHC",,r3   c                     t          |||z
  z  |z             dz
  }|dz
                      ||          }|                     |||          }t          j        ||k    ||          S rE   )r   r  rU   r,   r  )r/   ra   r  r  r{   r  r  s          r0   rb   zrandint_gen._ppf  sf    A$s*++a/T**yyT**x	5$///r3   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      (@rj  g333333)r,   rJ  )
r/   r  r  m2m1rl   drm   rn   ro   s
             r0   ru   zrandint_gen._stats  s|    D!!2:c??B2gmq GsQw$1s#qsSy13Br3   Nc                    t          j        |          j        dk    r0t          j        |          j        dk    rt          ||||          S |*t          j        ||          }t          j        ||          }t          j        t          t          |          t          j        t                    g          } |||          S )z=An array of *size* random integers >= ``low`` and < ``high``.r   r  N)otypes)	r,   rJ  r8   r   broadcast_to	vectorizer   r   r`  )r/   r  r  r8   r9   randints         r0   r:   zrandint_gen._rvs  s    :c??1$$D)9)9)>!)C)Cc4dCCCC
 /#t,,C?4..D,w|\BB')x}}o7 7 7wsD!!!r3   c                 &    t          ||z
            S r5   )r   r  s      r0   r|   zrandint_gen._entropy  s    4#:r3   re   )r   r   r   r   r1   r?   rC   rQ   rU   rb   ru   r:   r|   r   r3   r0   r  r    s        = =~M M MC C C  8 8 8
- - -0 0 0  " " " ""    r3   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 zipf, 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 )NrB   Fr   r   r+   r.   s    r0   r1   zzipf_gen._shape_info;      326{NCCDDr3   Nc                 0    |                     ||          S r)  )zipf)r/   rB   r8   r9   s       r0   r:   zzipf_gen._rvs>  s       ...r3   c                     |dk    S rE   r   r/   rB   s     r0   r?   zzipf_gen._argcheckA  s    1ur3   c                     |                     t          j                  }dt          j        |d          z  || z  z  }|S Nr   r   )r   r,   float64r   r	   )r/   rJ   rB   r  s       r0   rQ   zzipf_gen._pmfD  s;    HHRZ  7<1%%%A2-	r3   c                 N    t          ||dz   k    ||fd t          j                  S )Nr   c                 ^    t          j        | |z
  d          t          j        | d          z  S rE   )r   r	   )rB   r&   s     r0   ri  z zipf_gen._munp.<locals>.<lambda>M  s'    a!eQ//',q!2D2DD r3   r  )r/   r&   rB   s      r0   _munpzzipf_gen._munpJ  s0    AI1vDDF  	r3   re   )	r   r   r   r   r1   r:   r?   rQ   r  r   r3   r0   r  r    sr        ) )VE E E/ / / /        r3   r  r  zA Zipfc                 J    t          |d          t          || dz             z
  S )z"Generalized harmonic number, a > 1r   )r	   r&   rB   s     r0   _gen_harmonic_gt1r  T  s#     1::Q!$$r3   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 <= 1r  r   r   )r,   r8   r  
zeros_likefloatrI  )r&   rB   n_maxoutimasks         r0   _gen_harmonic_leq1r  Z  s    71:: F1IIE
-
'
'
'CYua5111 " "AvD			Qq!D'z\!				Jr3   c                 x    t          j        | |          \  } }t          |dk    | |ft          t                    S )zGeneralized harmonic numberr   r   f2)r,   r   r
   r  r  r  s     r0   _gen_harmonicr  g  sE    q!$$DAqa!eaV).@B B B Br3   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, zipfian
    >>> 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 )NrB   Fr   r'   r&   Tr   r+   r.   s    r0   r1   zzipfian_gen._shape_info  s<    326{MBB3q"&k>BBD 	Dr3   c                 \    |dk    |dk    z  |t          j        |t                    k    z  S )Nr   r  )r,   rJ  r`  r/   rB   r&   s      r0   r?   zzipfian_gen._argcheck  s.    Q1q5!Q"*Qc*B*B*B%BCCr3   c                 
    d|fS rE   r   r  s      r0   rC   zzipfian_gen._get_support  r   r3   c                 t    |                     t          j                  }dt          ||          z  || z  z  S r   )r   r,   r  r  r/   rJ   rB   r&   s       r0   rQ   zzipfian_gen._pmf  s5    HHRZ  ]1a(((1qb500r3   c                 D    t          ||          t          ||          z  S r5   r  r  s       r0   rU   zzipfian_gen._cdf  s!    Q""]1a%8%888r3   c                     |dz   }||z  t          ||          t          ||          z
  z  dz   ||z  t          ||          z  z  S rE   r  r  s       r0   rY   zzipfian_gen._sf  sU    EA}Q**]1a-@-@@AAEa4a+++- 	.r3   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   r  rv  r   r  )r/   rB   r&   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2rn   ro   s                 r0   ru   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r3   N)r   r   r   r   r1   r?   rC   rQ   rU   rY   ru   r   r3   r0   r  r  n  s        , ,\D D DD D D  1 1 19 9 9. . .         r3   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 )NrB   Fr   r   r+   r.   s    r0   r1   zdlaplace_gen._shape_info  r  r3   c                 h    t          |dz            t          | t          |          z            z  S Nrg   )r   r   abs)r/   rJ   rB   s      r0   rQ   zdlaplace_gen._pmf  s+    AcE{{S!c!ff----r3   c                 ^    t          |          }d }d }t          |dk    ||f||          S )Nc                 T    dt          | | z            t          |          dz   z  z
  S r  r  rJ   rB   s     r0   r   zdlaplace_gen._cdf.<locals>.f  s(    aR!VA
333r3   c                 R    t          || dz   z            t          |          dz   z  S rE   r  r  s     r0   r  zdlaplace_gen._cdf.<locals>.f2  s'    qAE{##s1vvz22r3   r   r  )r   r
   )r/   rI   rB   rJ   r   r  s         r0   rU   zdlaplace_gen._cdf  sL    !HH	4 	4 	4	3 	3 	3 !q&1a&A"5555r3   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   rU   )r/   ra   rB   constr{   r  s         r0   rb   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%>>>r3   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 )Nrg   r   r   g      $@r  rj  r   r  )r/   rB   ear  r{  s        r0   ru   zdlaplace_gen._stats  si    VVeRUQJeRU3r6\"_%B
23CQJO++r3   c                 f    |t          |          z  t          t          |dz                      z
  S r  )r   r   r   r  s     r0   r|   zdlaplace_gen._entropy  s)    477{Sae----r3   Nc                     t          j        t          j        |                      }|                    ||          }|                    ||          }||z
  S r)  )r,   r   rJ  r	  )r/   rB   r8   r9   probOfSuccessrI   ys          r0   r:   zdlaplace_gen._rvs  sY      2:a==.111""=t"<<""=t"<<1ur3   re   )r   r   r   r   r1   rQ   rU   rb   ru   r|   r:   r   r3   r0   r  r    s         ,E E E. . .	6 	6 	6? ? ?, , ,. . .     r3   r  dlaplacezA discrete Laplacianc                   J    e Zd ZdZd Zd ZddddZd Zd Zd	 Z	d
 Z
d ZdS )poisson_binom_genul  A Poisson Binomial discrete random variable.

    %(before_notes)s

    See Also
    --------
    binom

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

    .. math::

     f(k; p_1, p_2, ..., p_n) = \sum_{A \in F_k} \prod_{i \in A} p_i \prod_{j \in A^C} 1 - p_j

    where :math:`k \in \{0, 1, \dots, n-1, n\}`, :math:`F_k` is the set of all
    subsets of :math:`k` integers that can be selected :math:`\{0, 1, \dots, n-1, n\}`,
    and :math:`A^C` is the complement of a set :math:`A`.

    `poisson_binom` accepts a single array argument ``p`` for shape parameters
    :math:`0 ≤ p_i ≤ 1`, where the last axis corresponds with the index :math:`i` and
    any others are for batch dimensions. Broadcasting behaves according to the usual
    rules except that the last axis of ``p`` is ignored. Instances of this class do
    not support serialization/unserialization.

    %(after_notes)s

    References
    ----------
    .. [1] "Poisson binomial distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Poisson_binomial_distribution
    .. [2] Biscarri, William, Sihai Dave Zhao, and Robert J. Brunner. "A simple and
           fast method for computing the Poisson binomial distribution function".
           Computational Statistics & Data Analysis 122 (2018) 92-100.
           :doi:`10.1016/j.csda.2018.01.007`

    %(example)s

    c                     g S r5   r   r.   s    r0   r1   zpoisson_binom_gen._shape_infoG  s	     	r3   c                 t    t          j        |d          }d|k    |dk    z  }t          j        |d          S Nr   rw   r   )r,   stackall)r/   argsr(   condss       r0   r?   zpoisson_binom_gen._argcheckL  s=    HT"""aAF#ve!$$$$r3   Nr   c                *   t          j        |d          }||j        n)t          j        |          r|dfnt	          |          dz   }t          j        |j        |          }t                              |||                              d          S )Nr  rw   r   )r   r   )	r,   r  shapeisscalartuplebroadcast_shapesr   r:   rz   )r/   r8   r9   r!  r(   s        r0   r:   zpoisson_binom_gen._rvsQ  s    HT###  <[..Fq		E$KK$4F 	"17D11~~ad~FFJJPRJSSSr3   c                 $    dt          |          fS r   )len)r/   r!  s     r0   rC   zpoisson_binom_gen._get_support[  s    #d))|r3   c                     t          j        |                              t           j                  }t          j        |g|R  ^}}t          j        |t           j                  }t          ||d          S )Nr  rC  r,   
atleast_1dr   r  r   rJ  r  r!   r/   rJ   r!  s      r0   rQ   zpoisson_binom_gen._pmf^  e    M!##BH--&q04000Dz$bj111au---r3   c                     t          j        |                              t           j                  }t          j        |g|R  ^}}t          j        |t           j                  }t          ||d          S )Nr  r   r+  r-  s      r0   rU   zpoisson_binom_gen._cdfd  r.  r3   c                     t          j        |d          }t          j        |d          }t          j        |d|z
  z  d          }||d d fS r  )r,   r  rz   )r/   r!  kwdsr(   r   rm   s         r0   ru   zpoisson_binom_gen._statsj  sU    HT"""vaa   fQ!A#YQ'''c4&&r3   c                 "    t          | g|R i |S r5   )poisson_binomial_frozen)r/   r!  r1  s      r0   __call__zpoisson_binom_gen.__call__p  s     &t;d;;;d;;;r3   )r   r   r   r   r1   r?   r:   rC   rQ   rU   ru   r4  r   r3   r0   r  r    s        ' 'P  
% % %
  $$ T T T T T  . . .. . .' ' '< < < < <r3   r  poisson_binomzA Poisson binomialr(   )r   r!  shapesc                 P    t          t          j        |dd                    |d|fS Nr  r   r   r&  r,   moveaxis)r/   r(   locr8   s       r0   _parse_args_rvsr<  |  s'    QA&&''c477r3   rc   c                 P    t          t          j        |dd                    |d|fS r8  r9  )r/   r(   r;  rk   s       r0   _parse_args_statsr>    s'    QA&&''c7::r3   c                 N    t          t          j        |dd                    |dfS r8  r9  )r/   r(   r;  s      r0   _parse_argsr@    s%    QA&&''c11r3   c                       e Zd Zd ZddZdS )r3  c                    || _         || _         |j        di |                                | _        t
                              t          t                    | j        _        t                              t          t                    | j        _	        t                              t          t                    | j        _
         | j        j
        |i |\  }}} | j        j        | \  | _        | _        d S )Nr   )r!  r1  	__class___updated_ctor_paramdistr<  __get___pb_obj_pb_clsr>  r@  rC   rB   r   )r/   rE  r!  r1  r6  _s         r0   __init__z poisson_binomial_frozen.__init__  s    		 #DN@@T%=%=%?%?@@	 %4$;$;GW$M$M	!&7&?&?&Q&Q	# + 3 3GW E E	,ty,d;d;;1//8r3   NFc                 |     | j         j        | j        i | j        \  }}} | j         j        || j        ||||fi |S r5   )rE  r@  r!  r1  expect)	r/   funclbubconditionalr1  rB   r;  scales	            r0   rL  zpoisson_binomial_frozen.expect  sP    -	-tyFDIFF3  tydib"kRRTRRRr3   )NNNF)r   r   r   rJ  rL  r   r3   r0   r3  r3    s=        9 9 9S S S S S Sr3   r3  c                   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  r+   r.   s    r0   r1   zskellam_gen._shape_info  s<    5%!RVnEE5%!RVnEEG 	Gr3   Nc                 `    |}|                     ||          |                     ||          z
  S r5   r  )r/   r  r  r8   r9   r&   s         r0   r:   zskellam_gen._rvs  s7    $$S!,,$$S!,,- 	.r3   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  rN   	_ncx2_pdfr/   rI   r  r  pxs        r0   rQ   zskellam_gen._pmf  s    [h''' 	B 	B!a%-#q!A#w#>>q@-#q!A#w#>>q@B BB	B 	B 	B 	B 	B 	B 	B 	B 	B 	B 	B 	B 	B 	B 	B
 	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  rN   	_ncx2_cdfrX  s        r0   rU   zskellam_gen._cdf  s    !HH[h''' 	D 	D!a%-#r!tQsU;;cmAcE1ac7AcEBBBD DB	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	D 	s   ABBBc                 V    ||z
  }||z   }|t          |dz            z  }d|z  }||||fS )Nr   r   r   )r/   r  r  r   rm   rn   ro   s          r0   ru   zskellam_gen._stats  s@    SyCiD#NN"WS"b  r3   re   )	r   r   r   r   r1   r:   rQ   rU   ru   r   r3   r0   rS  rS    sq         :G G G. . . .
    ! ! ! ! !r3   rS  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   r+   r.   s    r0   r1   zyulesimon_gen._shape_info  s    7EArv;GGHHr3   Nc           	          |                     |          }|                     |          }t          | t          t          | |z                       z            }|S r5   )standard_exponentialr   r   r   )r/   rb  r8   r9   E1E2anss          r0   r:   zyulesimon_gen._rvs  sZ    ..t44..t44B3RC%K 0 0011122
r3   c                 8    |t          j        ||dz             z  S rE   r   r   r/   rI   rb  s      r0   rQ   zyulesimon_gen._pmf  s    w|Auqy1111r3   c                     |dk    S r   r   )r/   rb  s     r0   r?   zyulesimon_gen._argcheck  s    	r3   c                 R    t          |          t          j        ||dz             z   S rE   r   r   r   rj  s      r0   rL   zyulesimon_gen._logpmf  s#    5zzGN1eai8888r3   c                 >    d|t          j        ||dz             z  z
  S rE   ri  rj  s      r0   rU   zyulesimon_gen._cdf  s"    1w|Auqy11111r3   c                 8    |t          j        ||dz             z  S rE   ri  rj  s      r0   rY   zyulesimon_gen._sf  s    7<519----r3   c                 R    t          |          t          j        ||dz             z   S rE   rm  rj  s      r0   r  zyulesimon_gen._logsf   s#    1vvq%!)4444r3   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|dz  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   rg   r   r     1      )r,   r  r-   nanr   )r/   rb  rl   r  rn   ro   s         r0   ru   zyulesimon_gen._stats#  s\   Xeqj"&%519*=>>huqyaxECKEAI>#ABv  huz263//Xeai519ooQ6%519:MNf  Xeqj"&"--XeaiaiBMBJ$>$C$)UQY$7519$E$G Hf  Xeqj"&"--3Br3   re   )r   r   r   r   r1   r:   rQ   r?   rL   rU   rY   r  ru   r   r3   r0   r`  r`    s           BI I I   2 2 2  9 9 92 2 2. . .5 5 5    r3   r`  	yulesimon)r   rB   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   r+   r.   s    r0   r1   z_nchypergeom_gen._shape_infoB  sj    3q"&k=AA3q"&k=AA3q"&k=AA651bf+~FFH 	Hr3   c                 z    |||}}}||z
  }t          j        d||z
            }t          j        ||          }||fS r   r,  )	r/   r%  r&   r&  rz  r  r  x_minx_maxs	            r0   rC   z_nchypergeom_gen._get_supportH  sH    aq2V
1a"f%%
1b!!e|r3   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,   rJ  r   r`  )r/   r%  r&   r&  rz  cond1cond2cond3cond4cond5cond6s              r0   r?   z_nchypergeom_gen._argcheckO  s    z!}}bjmm1*Q--D!1!14#!#Q/#!#Q/#!#Q/qQQu}u$u,u4u<<r3   c                 J     t            fd            } |||||||          S )Nc                     t          j        |          }t                      }t          |
j                  } |||| |||          }	|	                    |          }	|	S r5   )r,   prodr    getattrrvs_namereshape)r%  r&   r&  rz  r8   r9   lengthurnrv_genrd  r/   s             r0   re  z$_nchypergeom_gen._rvs.<locals>._rvs1\  s[    WT]]F#%%CS$-00F&Aq$==C++d##CJr3   r   rf  )r/   r%  r&   r&  rz  r8   r9   re  s   `       r0   r:   z_nchypergeom_gen._rvsZ  sF    	#	 	 	 	 
$	#	 uQ1dLIIIIr3   c                      t          j        |||||          \  }}}}}|j        dk    rt          j        |          S t           j         fd            } ||||||          S )Nr   c                 `                         ||||d          }|                    |           S Ng-q=)rE  probability)rI   r%  r&   r&  rz  r  r/   s         r0   _pmf1z$_nchypergeom_gen._pmf.<locals>._pmf1m  s.    ))Aq!T511C??1%%%r3   )r,   r   r8   
empty_liker  )r/   rI   r%  r&   r&  rz  r  s   `      r0   rQ   z_nchypergeom_gen._pmfg  s    .q!Q4@@1aD6Q;;=###		& 	& 	& 	& 
	& uQ1a&&&r3   c                 ~     t           j         fd            }d|v sd|v r |||||          nd\  }}d\  }	}
|||	|
fS )Nc                 ^                         ||| |d          }|                                S r  )rE  rk   )r%  r&   r&  rz  r  r/   s        r0   	_moments1z*_nchypergeom_gen._stats.<locals>._moments1v  s*    ))Aq!T511C;;== r3   r@  vre   )r,   r  )r/   r%  r&   r&  rz  rk   r  r@  r  rf   rJ   s   `          r0   ru   z_nchypergeom_gen._statst  ss    		! 	! 	! 	! 
	! .1G^^sg~~		!Q4(((! 	11!Qzr3   re   )r   r   r   r   r  rE  r1   rC   r?   r:   rQ   ru   r   r3   r0   rx  rx  8  s          HDH H H  	= 	= 	=J J J J' ' '
 
 
 
 
r3   rx  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   r  r   rE  r   r3   r0   r  r    s'        G GR H%DDDr3   r  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   r  r   rE  r   r3   r0   r  r    s'        G GR H'DDDr3   r  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)r   N)r   rc   )r   )j	functoolsr   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   
_biasedurnr   r   r    _stats_pythranr!   scipy.special._ufuncs_ufuncsrN   r#   r   r   r   r   r   r   r   r   r  r  r   r#  rQ  rS  rm  ro  r|  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r-   r  r  r5  r<  r>  r@  rG  rH  rF  r3  rS  r^  r`  rv  rx  r  r  r  r  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r   r3   r0   <module>r     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    8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8, , , , , , , , , , + * * * * * # # # # # # # # #]* ]* ]* ]* ]* ]* ]* ]*@ 		w># ># ># ># >#I ># ># >#B MAK000	O O O O OK O O Od M{+++	r
 r
 r
 r
 r
 r
 r
 r
j 
	"	"	"Z Z Z Z Z[ Z Z Zz ^...
N7 N7 N7 N7 N7{ N7 N7 N7b x!&=999X X X X XK X X Xv M{+++	\ \ \ \ \[ \ \ \~ ^...
5 5 5 5 5 5 5 5p 
ah	A	A	A? ? ? ? ?+ ? ? ?D +9{
;
;
;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	t t t t t+ t t tn +9 0) * * *
? ? ? ? ?{ ? ? ?D x!&8444% % %
 
 
B B BV  V  V  V  V + V  V  V r +	K
@
@
@M M M M M; M M M` <26''2HJ J JS< S< S< S< S< S< S< S<l "!AU),. . .8 8 8 8; ; ; ;2 2 2 2
 !"3  / 7 7 I I "3";";GW"M"M '//AA S S S S S0 S S S0<! <! <! <! <!+ <! <! <!~ +i+
F
F
FL L L L LK L L L^ M{a000	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 
)
)r3   