
    doi3                    `   S SK Jr  S SKJr  S SKJrJrJrJr	  S SK
Js  Jr  S SKJr  S SKJs  Jr  S SKJr  S SKJrJrJrJrJrJrJrJrJrJr  S SKr SS	K!J"r"J#r#J$r$J%r%J&r&J'r'  SS
K(J)r)J*r*J+r+  SSK,J-r-   " S S\"5      r.\." SS9r/ " S S\.5      r0\0" SSS9r1 " S S\"5      r2\2" SS9r3 " S S\"5      r4\4" SS9r5 " S S\"5      r6\6" SS9r7 " S S\"5      r8\8" SSS S!9r9 " S" S#\"5      r:\:" S$S9r; " S% S&\"5      r<\<" S'S9r= " S( S)\"5      r>\>" SS*S+S!9r? " S, S-\"5      r@\@" S.S/S09rA " S1 S2\"5      rB\B" S S3S4S!9rC " S5 S6\"5      rD\D" S7S S8S99rE " S: S;\"5      rF\F" S<S=S09rG " S> S?\"5      rH\H" SS@SAS!9rI " SB SC\"5      rJ\J" SSDSES!9rK " SF SG\"5      rL\L" \ R                  * SHSIS!9rN " SJ SK\"5      rO\O" SLSMSNSO9rPSgSP jrQShSQ jrRSiSR jrS\P\OsrTrU\QR                  \T\U5      \PlQ        \RR                  \T\U5      \PlR        \SR                  \T\U5      \PlS         " SS ST\'5      rW " SU SV\"5      rX\X" \ R                  * SWSXS!9rY " SY SZ\"5      rZ\Z" S[SS\9r[ " S] S^\"5      r\ " S_ S`\\5      r]\]" SaSbS09r^ " Sc Sd\\5      r_\_" SeSfS09r`\a" \b" 5       R                  5       R                  5       5      re\#" \e\"5      u  rfrg\f\g-   rhg)j    )partial)special)entr	logsumexpbetalngammalnN)rng_integers)interp1d)
floorceillogexpsqrtlog1pexpm1tanhcoshsinh   )rv_discreteget_distribution_names_vectorize_rvs_over_shapes
_ShapeInfo_isintegralrv_discrete_frozen)_PyFishersNCHypergeometric_PyWalleniusNCHypergeometric_PyStochasticLib3)_poisson_binomc                   h    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rS rSS jrS rSrg)	binom_gen   a  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                 Z    [        SSS[        R                  4S5      [        SSSS5      /$ 	NnTr   TFpFr   r   TTr   npinfselfs    X/var/www/html/land-ocr/venv/lib/python3.13/site-packages/scipy/stats/_discrete_distns.py_shape_infobinom_gen._shape_info@   0    3q"&&k=A3v|<> 	>    Nc                 &    UR                  XU5      $ N)binomialr.   r%   r'   sizerandom_states        r/   _rvsbinom_gen._rvsD   s    $$Q400r3   c                 <    US:  [        U5      -  US:  -  US:*  -  $ Nr   r   r   r.   r%   r'   s      r/   	_argcheckbinom_gen._argcheckG   s'    Q+a.(AF3qAv>>r3   c                     U R                   U4$ r5   ar?   s      r/   _get_supportbinom_gen._get_supportJ   s    vvqyr3   c                     [        U5      n[        US-   5      [        US-   5      [        X$-
  S-   5      -   -
  nU[        R                  " XC5      -   [        R                  " X$-
  U* 5      -   $ Nr   )r   gamlnr   xlogyxlog1py)r.   xr%   r'   kcombilns         r/   _logpmfbinom_gen._logpmfM   s\    !H1:qseACEl!:;q,,wqsQB/GGGr3   c                 0    [         R                  " XU5      $ r5   )scu
_binom_pmfr.   rL   r%   r'   s       r/   _pmfbinom_gen._pmfR   s    ~~aA&&r3   c                 F    [        U5      n[        R                  " XBU5      $ r5   )r   rR   
_binom_cdfr.   rL   r%   r'   rM   s        r/   _cdfbinom_gen._cdfV       !H~~aA&&r3   c                 F    [        U5      n[        R                  " XBU5      $ r5   )r   rR   	_binom_sfrY   s        r/   _sfbinom_gen._sfZ   s    !H}}Q1%%r3   c                 0    [         R                  " XU5      $ r5   )rR   
_binom_isfrT   s       r/   _isfbinom_gen._isf^       ~~aA&&r3   c                 0    [         R                  " XU5      $ r5   )rR   
_binom_ppfr.   qr%   r'   s       r/   _ppfbinom_gen._ppfa   re   r3   c                    X-  nXA[         R                  " U5      -  -
  nSu  pgSU;   aS  U[         R                  " U5      -
  n[         R                  " X-  5      n	[         R                  " U	5      n
SU-  U	-  nX-
  nSU;   a<  U[         R                  " U5      -
  nX-  n[         R                  " U5      n
SU-  nX-
  nXEXg4$ )NNNs       @rM         @)r+   squarer   
reciprocal)r.   r%   r'   momentsmuvarg1g2pqnpq_sqrtt1t2npqs                r/   _statsbinom_gen._statsd   s    Uryy|##'>RYYq\!BwwqvHx(B'X%BB'>RYYq\!B&Cs#BQBBr3   c                     [         R                  SUS-    nU R                  X1U5      n[         R                  " [	        U5      SS9$ )Nr   r   axis)r+   r_rU   sumr   )r.   r%   r'   rM   valss        r/   _entropybinom_gen._entropyv   s:    EE!AENyyq!vvd4jq))r3    rm   mv__name__
__module____qualname____firstlineno____doc__r0   r:   r@   rE   rO   rU   rZ   r_   rc   rj   r}   r   __static_attributes__r   r3   r/   r!   r!      sE    "F>1?H
''&''$*r3   r!   binom)namec                   d    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rS rS rS rSrg)bernoulli_gen   a  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                      [        SSSS5      /$ Nr'   Fr(   r)   r   r-   s    r/   r0   bernoulli_gen._shape_info       3v|<==r3   Nc                 .    [         R                  U SXUS9$ )Nr   r8   r9   )r!   r:   r.   r'   r8   r9   s       r/   r:   bernoulli_gen._rvs   s    ~~dAq,~OOr3   c                     US:  US:*  -  $ r=   r   r.   r'   s     r/   r@   bernoulli_gen._argcheck   s    Q16""r3   c                 2    U R                   U R                  4$ r5   )rD   br   s     r/   rE   bernoulli_gen._get_support   s    vvtvv~r3   c                 0    [         R                  USU5      $ rH   )r   rO   r.   rL   r'   s      r/   rO   bernoulli_gen._logpmf   s    }}Q1%%r3   c                 0    [         R                  USU5      $ rH   )r   rU   r   s      r/   rU   bernoulli_gen._pmf   s     zz!Q""r3   c                 0    [         R                  USU5      $ rH   )r   rZ   r   s      r/   rZ   bernoulli_gen._cdf       zz!Q""r3   c                 0    [         R                  USU5      $ rH   )r   r_   r   s      r/   r_   bernoulli_gen._sf   s    yyAq!!r3   c                 0    [         R                  USU5      $ rH   )r   rc   r   s      r/   rc   bernoulli_gen._isf   r   r3   c                 0    [         R                  USU5      $ rH   )r   rj   )r.   ri   r'   s      r/   rj   bernoulli_gen._ppf   r   r3   c                 .    [         R                  SU5      $ rH   )r   r}   r   s     r/   r}   bernoulli_gen._stats   s    ||Aq!!r3   c                 6    [        U5      [        SU-
  5      -   $ rH   )r   r   s     r/   r   bernoulli_gen._entropy   s    Awac""r3   r   rm   r   r   r3   r/   r   r      sD    0>P#&#
#"##"#r3   r   	bernoulli)r   r   c                   J    \ rS rSrSrS rSS jrS rS rS r	S	 r
SS
 jrSrg)betabinom_gen   a  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.

%(after_notes)s

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

.. versionadded:: 1.4.0

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

%(example)s

c                     [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ 	Nr%   Tr   r&   rD   FFFr   r*   r-   s    r/   r0   betabinom_gen._shape_info   P    3q"&&k=A3266{NC3266{NCE 	Er3   Nc                 J    UR                  X#U5      nUR                  XU5      $ r5   )betar6   r.   r%   rD   r   r8   r9   r'   s          r/   r:   betabinom_gen._rvs   s'    aD)$$Q400r3   c                 
    SU4$ Nr   r   r.   r%   rD   r   s       r/   rE   betabinom_gen._get_support       !tr3   c                 <    US:  [        U5      -  US:  -  US:  -  $ r   r>   r   s       r/   r@   betabinom_gen._argcheck   '    Q+a.(AE2a!e<<r3   c                     [        U5      n[        US-   5      * [        X%-
  S-   US-   5      -
  nU[        XS-   X%-
  U-   5      -   [        X45      -
  $ rH   )r   r   r   r.   rL   r%   rD   r   rM   rN   s          r/   rO   betabinom_gen._logpmf   sS    !Hq1u:+quqy!a% 88quqy11F1L@@r3   c                 8    [        U R                  XX45      5      $ r5   r   rO   r.   rL   r%   rD   r   s        r/   rU   betabinom_gen._pmf       4<<a+,,r3   c                 (   X"U-   -  nSU-
  nX-  nXU-   U-   -  U-  U-  X#-   S-   -  nSu  pSU;   a/  S[        U5      -  n	XU-   SU-  -   X2-
  -  -  n	XU-   S-   X#-   -  -  n	SU;   a  X#-   R                  UR                  5      n
XU-   S-
  SU-  -   -  n
U
SU-  U-  US-
  -  -  n
U
SUS-  -  -  n
U
SU-  U-  U-  SU-
  -  -  n
U
S	U-  U-  US-  -  -  n
XU-   S-  SU-   U-   -  -  n
XU-  U-  X#-   S-   -  X#-   S-   -  X#-   U-   -  -  n
U
S-  n
XxX4$ )
Nr   rm   rn         ?   rM            )r   astypedtype)r.   r%   rD   r   rs   e_pe_qrt   ru   rv   rw   s              r/   r}   betabinom_gen._stats   s   q5k#gWq519o#c)QUQY7'>tCyBq51q5=QU++Bq519''B'>%		*Bq519q1u$%B!a%!)q1u%%B!a1f*B!c'A+/QU++B"s(S.16))Bq5Q,!a%!),,Bq519	*aeai8AEAIFGB!GBr3   r   rm   r   )r   r   r   r   r   r0   r:   rE   r@   rO   rU   r}   r   r   r3   r/   r   r      s-    "FE
1=A
-r3   r   	betabinomc                   ^    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rS rS rSrg)
nbinom_geni  a?  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                 Z    [        SSS[        R                  4S5      [        SSSS5      /$ r$   r*   r-   s    r/   r0   nbinom_gen._shape_infoS  r2   r3   Nc                 &    UR                  XU5      $ r5   )negative_binomialr7   s        r/   r:   nbinom_gen._rvsW  s    --aD99r3   c                 $    US:  US:  -  US:*  -  $ r=   r   r?   s      r/   r@   nbinom_gen._argcheckZ  s    A!a% AF++r3   c                 0    [         R                  " XU5      $ r5   )rR   _nbinom_pmfrT   s       r/   rU   nbinom_gen._pmf]  s    qQ''r3   c                     [        X!-   5      [        US-   5      -
  [        U5      -
  nXB[        U5      -  -   [        R                  " X* 5      -   $ rH   )rI   r   r   rK   )r.   rL   r%   r'   coeffs        r/   rO   nbinom_gen._logpmfa  sD    ac
U1Q3Z'%(2Qx'//!R"888r3   c                 F    [        U5      n[        R                  " XBU5      $ r5   )r   rR   _nbinom_cdfrY   s        r/   rZ   nbinom_gen._cdfe  s    !HqQ''r3   c                 B   [        U5      n[        R                  " XBU5      u  pBnU R                  XBU5      nUS:  nS nUn[        R                  " SS9   U" XF   X&   X6   5      X'   [        R
                  " XV)    5      X) '   S S S 5        U$ ! , (       d  f       U$ = f)N      ?c                 h    [         R                  " [        R                  " U S-   USU-
  5      * 5      $ rH   )r+   r   r   betainc)rM   r%   r'   s      r/   f1nbinom_gen._logcdf.<locals>.f1n  s)    88W__QUAq1u==>>r3   ignore)divide)r   r+   broadcast_arraysrZ   errstater   )	r.   rL   r%   r'   rM   cdfcondr   logcdfs	            r/   _logcdfnbinom_gen._logcdfi  s    !H%%aA.aiia Sy	? [[)agqw8FLFF3u:.F5M *  *) s   /B
Bc                 F    [        U5      n[        R                  " XBU5      $ r5   )r   rR   
_nbinom_sfrY   s        r/   r_   nbinom_gen._sfx  r\   r3   c                     [         R                  " SS9   [        R                  " XU5      sS S S 5        $ ! , (       d  f       g = fNr   over)r+   r   rR   _nbinom_isfrT   s       r/   rc   nbinom_gen._isf|  (    [[h'??1+ (''	   6
Ac                     [         R                  " SS9   [        R                  " XU5      sS S S 5        $ ! , (       d  f       g = fr  )r+   r   rR   _nbinom_ppfrh   s       r/   rj   nbinom_gen._ppf  r
  r  c                     [         R                  " X5      [         R                  " X5      [         R                  " X5      [         R                  " X5      4$ r5   )rR   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessr?   s      r/   r}   nbinom_gen._stats  sD    Q"  &  &''-	
 	
r3   r   rm   )r   r   r   r   r   r0   r:   r@   rU   rO   rZ   r   r_   rc   rj   r}   r   r   r3   r/   r   r     s?    9t>:,(9(',,
r3   r   nbinomc                   D    \ rS rSrSrS rSS jrS rS rS r	SS	 jr
S
rg)betanbinom_geni  a  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.

%(after_notes)s

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

.. versionadded:: 1.12.0

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

%(example)s

c                     [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ r   r*   r-   s    r/   r0   betanbinom_gen._shape_info  r   r3   Nc                 J    UR                  X#U5      nUR                  XU5      $ r5   )r   r   r   s          r/   r:   betanbinom_gen._rvs  s'    aD)--aD99r3   c                 <    US:  [        U5      -  US:  -  US:  -  $ r   r>   r   s       r/   r@   betanbinom_gen._argcheck  r   r3   c                     [        U5      n[        R                  " X%-   5      * [        X%S-   5      -
  nU[        X2-   XE-   5      -   [        X45      -
  $ rH   )r   r+   r   r   r   s          r/   rO   betanbinom_gen._logpmf  sI    !H66!%=.6!U#33qu--q<<r3   c                 8    [        U R                  XX45      5      $ r5   r   r   s        r/   rU   betanbinom_gen._pmf  r   r3   c                    S n[         R                  " US:  XU4U[        R                  S9nS n[         R                  " US:  XU4U[        R                  S9nSu  pS n
SU;   a*  [         R                  " US	:  XU4U
[        R                  S9nS
 nSU;   a*  [         R                  " US:  XU4U[        R                  S9n	XgX4$ )Nc                     X-  US-
  -  $ Nr   r   r%   rD   r   s      r/   mean#betanbinom_gen._stats.<locals>.mean  s    5AF##r3   r   
fill_valuec                 H    X-  X-   S-
  -  X-   S-
  -  US-
  US-
  S-  -  -  $ )Nr   ro   r   r%  s      r/   ru   "betanbinom_gen._stats.<locals>.var  s;    EQURZ(AEBJ7B1r6B,.0 1r3   r   rm   c                     SU -  U-   S-
  SU-  U-   S-
  -  US-
  -  [        X-  X-   S-
  -  X!-   S-
  -  US-
  -  5      -  $ )Nr   r         @ro   r   r%  s      r/   skew#betanbinom_gen._stats.<locals>.skew  sf    UQY^A	B72v!%aequrz&:aebj&I2v' "   !r3   rn   r   c                 t   US-
  nUS-
  S-  US-  USU-  S-
  -  -   SUS-
  -  U-  -   -  SU S-  -  US-   US-  -  US-   US-
  -  U-  -   SUS-
  S-  -  -   -  -   SUS-
  -  U -  US-   US-  -  US-   US-
  -  U-  -   SUS-
  S-  -  -   -  -   nUS	-
  US-
  -  U-  U -  X-   S-
  -  X-   S-
  -  nX4-  U-  S-
  $ )
Nro   r   r   rp   r-        @r   r   g      @r   )r%   rD   r   termterm_2denominators         r/   kurtosis'betanbinom_gen._stats.<locals>.kurtosis  sH   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   rM      xpxapply_wherer+   r,   )r.   r%   rD   r   rs   r&  rt   ru   rv   rw   r/  r6  s               r/   r}   betanbinom_gen._stats  s    	$__QUQ1ItG	1 ooa!eaAYG	! '>Qq	4BFFKB	4 '>Qq	8OBr3   r   rm   r   )r   r   r   r   r   r0   r:   r@   rO   rU   r}   r   r   r3   r/   r  r    s'    #HE
:==
-!r3   r  
betanbinomc                   ^    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rS rS rSrg)geom_geni  a  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                      [        SSSS5      /$ r   r   r-   s    r/   r0   geom_gen._shape_info  r   r3   Nc                     UR                  XS9n[        R                  " UR                  5      R                  n[        R
                  " US:  XT5      $ )Nr8   r   )	geometricr+   iinfor   maxwhere)r.   r'   r8   r9   resmax_ints         r/   r:   geom_gen._rvs  sD    $$Q$2 ((399%))xxa..r3   c                     US:*  US:  -  $ Nr   r   r   r   s     r/   r@   geom_gen._argcheck  s    Q1q5!!r3   c                 B    [         R                  " SU-
  US-
  5      U-  $ rH   )r+   powerr.   rM   r'   s      r/   rU   geom_gen._pmf  s     xx!QqS!A%%r3   c                 P    [         R                  " US-
  U* 5      [        U5      -   $ rH   )r   rK   r   rP  s      r/   rO   geom_gen._logpmf"  s"    q1uqb)CF22r3   c                 J    [        U5      n[        [        U* 5      U-  5      * $ r5   )r   r   r   r.   rL   r'   rM   s       r/   rZ   geom_gen._cdf%  s#    !HeQBik"""r3   c                 L    [         R                  " U R                  X5      5      $ r5   )r+   r   _logsfr   s      r/   r_   geom_gen._sf)  s    vvdkk!'((r3   c                 6    [        U5      nU[        U* 5      -  $ r5   )r   r   rU  s       r/   rX  geom_gen._logsf,  s    !Hr{r3   c                     [        [        U* 5      [        U* 5      -  5      nU R                  US-
  U5      n[        R                  " XA:  US:  -  US-
  U5      $ rL  )r   r   rZ   r+   rG  )r.   ri   r'   r   temps        r/   rj   geom_gen._ppf0  sS    E1"Iqb	)*yya#xxtax0$q&$??r3   c                     SU-  nSU-
  nX1-  U-  nSU-
  [        U5      -  n[        R                  " / SQU5      SU-
  -  nX$XV4$ )Nr   ro   )r   ir   )r   r+   polyval)r.   r'   rt   qrru   rv   rw   s          r/   r}   geom_gen._stats5  sT    UUfqj!etBxZZ
A&A.r3   c                 r    [         R                  " U5      * [         R                  " U* 5      SU-
  -  U-  -
  $ r$  )r+   r   r   r   s     r/   r   geom_gen._entropy=  s/    q	zBHHaRLCE2Q666r3   r   rm   )r   r   r   r   r   r0   r:   r@   rU   rO   rZ   r_   rX  rj   r}   r   r   r   r3   r/   r?  r?    s@    B>/"&3#)@
7r3   r?  geomzA geometric)rD   r   longnamec                   d    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rS rS rS rSrg)hypergeom_geniD  a?  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                     [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ )NMTr   r&   r%   Nr*   r-   s    r/   r0   hypergeom_gen._shape_info  P    3q"&&k=A3q"&&k=A3q"&&k=AC 	Cr3   Nc                 (    UR                  X!U-
  X4S9$ NrC  )hypergeometric)r.   rj  r%   rk  r8   r9   s         r/   r:   hypergeom_gen._rvs  s    **1c1*@@r3   c                 f    [         R                  " X1U-
  -
  S5      [         R                  " X#5      4$ r   r+   maximumminimum)r.   rj  r%   rk  s       r/   rE   hypergeom_gen._get_support  s'    zz!qS'1%rzz!'777r3   c                     US:  US:  -  US:  -  nXBU:*  X1:*  -  -  nU[        U5      [        U5      -  [        U5      -  -  nU$ r   r>   )r.   rj  r%   rk  r   s        r/   r@   hypergeom_gen._argcheck  sU    A!q&!Q!V,aAF##AQ/+a.@@r3   c                     X#peXV-
  n[        US-   S5      [        US-   S5      -   [        XT-
  S-   US-   5      -   [        US-   Xa-
  S-   5      -
  [        XA-
  S-   Xt-
  U-   S-   5      -
  [        US-   S5      -
  nU$ rH   r   )	r.   rM   rj  r%   rk  totgoodbadresults	            r/   rO   hypergeom_gen._logpmf  s    Tja#fSUA&66a19MM1dfQh'(*0Qa	*BCQ"# r3   c                 0    [         R                  " XXB5      $ r5   )rR   _hypergeom_pmfr.   rM   rj  r%   rk  s        r/   rU   hypergeom_gen._pmf      !!!--r3   c                 0    [         R                  " XXB5      $ r5   )rR   _hypergeom_cdfr  s        r/   rZ   hypergeom_gen._cdf  r  r3   c                 n   SU-  SU-  SU-  p2nX-
  nXS-   -  SU-  X-
  -  -
  SU-  U-  -
  nXQS-
  U-  U-  -  nUSU-  U-  X-
  -  U-  SU-  S-
  -  -  nXRU-  X-
  -  U-  US-
  -  US-
  -  -  n[         R                  " X#U5      [         R                  " X#U5      [         R                  " X#U5      U4$ )Nr   r   rp   r2  r   ro   r-  )rR   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r.   rj  r%   rk  mrw   s         r/   r}   hypergeom_gen._stats  s    q&"q&"q&aE a%[26QU++b1fqj8
1ukAo
b1fqjAE"Q&"q&1*55
!equo!QV,B77a(##A!,##A!,	
 	
r3   c                     [         R                  X1U-
  -
  [        X#5      S-    nU R                  XAX#5      n[         R                  " [        U5      SS9$ )Nr   r   r   )r+   r   minpmfr   r   )r.   rj  r%   rk  rM   r   s         r/   r   hypergeom_gen._entropy  sE    EE!1u+c!i!m,xxa#vvd4jq))r3   c                 0    [         R                  " XXB5      $ r5   )rR   _hypergeom_sfr  s        r/   r_   hypergeom_gen._sf  s      q,,r3   c                    / n[        [        R                  " XX45      6  H  u  pgpUS-   US-   -  US-
  U	S-
  -  :  a6  UR                  [	        [        U R                  XgX5      5      * 5      5        MT  [        R                  " US-   U	S-   5      n
UR                  [        U R                  XX5      5      5        M     [        R                  " U5      $ )Nr   r   )zipr+   r   appendr   r   r   aranger   rO   asarrayr.   rM   rj  r%   rk  rH  quantr{  r|  drawk2s              r/   rX  hypergeom_gen._logsf  s    &)2+>+>qQ+J&K"Ec	*dSjTCZ-HH

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

9T\\"4%FGH 'L zz#r3   c                    / n[        [        R                  " XX45      6  H  u  pgpUS-   US-   -  US-
  U	S-
  -  :  a6  UR                  [	        [        U R                  XgX5      5      * 5      5        MT  [        R                  " SUS-   5      n
UR                  [        U R                  XX5      5      5        M     [        R                  " U5      $ )Nr   r   r   )r  r+   r   r  r   r   logsfr  r   rO   r  r  s              r/   r   hypergeom_gen._logcdf  s    &)2+>+>qQ+J&K"Ec	*dSjTCZ-HH

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

9T\\"4%FGH 'L zz#r3   r   rm   )r   r   r   r   r   r0   r:   rE   r@   rO   rU   rZ   r}   r   r_   rX  r   r   r   r3   r/   rh  rh  D  sG    HRC
A8..
"*
-

r3   rh  	hypergeomc                   F    \ rS rSrSrS rS rS rSS jrS r	S	 r
S
 rSrg)nhypergeom_geni  a>  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                     [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ )Nrj  Tr   r&   r%   rr*   r-   s    r/   r0   nhypergeom_gen._shape_infoG  rm  r3   c                 
    SU4$ r   r   )r.   rj  r%   r  s       r/   rE   nhypergeom_gen._get_supportL  r   r3   c                     US:  X!:*  -  US:  -  X1U-
  :*  -  nU[        U5      [        U5      -  [        U5      -  -  nU$ r   r>   )r.   rj  r%   r  r   s        r/   r@   nhypergeom_gen._argcheckO  sK    Q16"a1f-c:AQ/+a.@@r3   Nc                 2   ^  [         U 4S j5       nU" XX4US9$ )Nc                   > TR                  XU5      u  pV[        R                  " XVS-   5      nTR                  XpX5      n[	        XSSS9n	U	" UR                  US95      R                  [        5      n
Uc  U
R                  5       $ U
$ )Nr   nextextrapolate)kindr)  rC  )	supportr+   r  r   r
   uniformr   intitem)rj  r%   r  r8   r9   rD   r   ksr   ppfrvsr.   s              r/   _rvs1"nhypergeom_gen._rvs.<locals>._rvs1V  s     <<a(DA1c"B((2!'C3MJCl***56==cBC|xxz!Jr3   r   r   )r.   rj  r%   r  r8   r9   r  s   `      r/   r:   nhypergeom_gen._rvsT  s&    	#		 
$		 Q1lCCr3   c                 H    [         R                  " US:g  US:g  -  XX44S SS9$ )Nr   c                     [        U S-   U5      * [        X-   S5      -   [        X -
  S-   X-
  U-
  S-   5      -
  [        X-
  U -
  S-   S5      -   [        US-   X-
  S-   5      -   [        US-   S5      -
  $ rH   rz  )rM   rj  r%   r  s       r/   <lambda>(nhypergeom_gen._logpmf.<locals>.<lambda>g  s    1a.6!#q>1!#a%Qq)*,213q57A,>?!A#qs1u%&(.qsA7r3           r(  )r:  r;  r.   rM   rj  r%   r  s        r/   rO   nhypergeom_gen._logpmfd  s3    !VQ!8  	r3   c                 8    [        U R                  XX45      5      $ r5   r   r  s        r/   rU   nhypergeom_gen._pmfm  s     4<<a+,,r3   c                     SU-  SU-  SU-  p2nX2-  X-
  S-   -  nX1S-   -  U-  X-
  S-   X-
  S-   -  -  SX1U-
  S-   -  -
  -  nSu  pgXEXg4$ )Nr   r   r   rm   r   )r.   rj  r%   r  rt   ru   rv   rw   s           r/   r}   nhypergeom_gen._statsr  sv     Q$1bdaSACE]1gaiACEACE?+q1!A;? r3   r   rm   )r   r   r   r   r   r0   rE   r@   r:   rO   rU   r}   r   r   r3   r/   r  r    s.    bHC

D -
r3   r  
nhypergeomc                   @    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
rg)
logser_geni  a  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                      [        SSSS5      /$ r   r   r-   s    r/   r0   logser_gen._shape_info  r   r3   Nc                      UR                  XS9$ ro  )	logseriesr   s       r/   r:   logser_gen._rvs  s     %%a%33r3   c                     US:  US:  -  $ r=   r   r   s     r/   r@   logser_gen._argcheck  s    A!a%  r3   c                 l    [         R                  " X!5      * S-  U-  [        R                  " U* 5      -  $ r$  )r+   rO  r   r   rP  s      r/   rU   logser_gen._pmf  s,    $q(7==!+<<<r3   c                     Sn[         R                  " US-   X25      * [         R                  " US-   U5      -  [        R                  " U* 5      -  $ )Ng0.++r   )r   r   r   r+   r   )r.   rM   r'   tinys       r/   r_   logser_gen._sf  sF    
 !T--QqS$0GG"((TUSU,VVr3   c                    [         R                  " U* 5      nXS-
  -  U-  nU* U-  US-
  S-  -  nXCU-  -
  nU* U-  SU-   -  SU-
  S-  -  nUSU-  U-  -
  SUS-  -  -   nU[        R                  " US5      -  nU* U-  SUS-
  S-  -  SU-  US-
  S-  -  -
  SU-  U-  US-
  S-  -  -   -  n	U	SU-  U-  -
  SU-  U-  U-  -   SUS-  -  -
  n
XS-  -  S-
  nX5X4$ )	Nr   r   r         ?r   r   r8  r-  )r   r   r+   rO  )r.   r'   r  rt   mu2pru   mu3pmu3rv   mu4pmu4rw   s               r/   r}   logser_gen._stats  s4   MM1"c']QrAvS1$UlrAvQ37Q,.QrT$Y2q5(288C%%rAv1Q3(NQqSAEA:--!A1q0@@BQtVBY42-"a%76\Cr3   r   rm   )r   r   r   r   r   r0   r:   r@   rU   r_   r}   r   r   r3   r/   r  r    s&    0>4
!=Wr3   r  logserzA logarithmicc                   R    \ rS rSrSrS rS rSS jrS rS r	S	 r
S
 rS rS rSrg)poisson_geni  ag  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                 @    [        SSS[        R                  4S5      /$ )Nrt   Fr   r&   r*   r-   s    r/   r0   poisson_gen._shape_info  s    4BFF]CDDr3   c                     US:  $ r   r   )r.   rt   s     r/   r@   poisson_gen._argcheck  s    Qwr3   Nc                 $    UR                  X5      $ r5   poisson)r.   rt   r8   r9   s       r/   r:   poisson_gen._rvs  s    ##B--r3   c                 V    [         R                  " X5      [        US-   5      -
  U-
  nU$ rH   )r   rJ   rI   )r.   rM   rt   Pks       r/   rO   poisson_gen._logpmf  s'    ]]1!E!a%L025	r3   c                 6    [        U R                  X5      5      $ r5   r   )r.   rM   rt   s      r/   rU   poisson_gen._pmf  s    4<<&''r3   c                 D    [        U5      n[        R                  " X25      $ r5   )r   r   pdtrr.   rL   rt   rM   s       r/   rZ   poisson_gen._cdf  s    !H||A""r3   c                 D    [        U5      n[        R                  " X25      $ r5   )r   r   pdtrcr  s       r/   r_   poisson_gen._sf  s    !H}}Q##r3   c                     [        [        R                  " X5      5      n[        R                  " US-
  S5      n[        R
                  " XB5      n[        R                  " XQ:  XC5      $ rL  )r   r   pdtrikr+   rt  r  rG  )r.   ri   rt   r   vals1r]  s         r/   rj   poisson_gen._ppf  sJ    GNN1)*

4!8Q'||E&xx	5//r3   c                     Un[         R                  " U5      nUS:  n[        R                  " XCS [         R                  S9n[        R                  " XCS [         R                  S9nXXV4$ )Nr   c                     [        SU -  5      $ r$  r.  rL   s    r/   r  $poisson_gen._stats.<locals>.<lambda>  s    SUr3   r(  c                     SU -  $ r$  r   r  s    r/   r  r    s    Ar3   )r+   r  r:  r;  r,   )r.   rt   ru   tmp
mu_nonzerorv   rw   s          r/   r}   poisson_gen._stats   sW    jjn1W
__Z.CPRPVPVW__Zo"&&Qr3   r   rm   )r   r   r   r   r   r0   r@   r:   rO   rU   rZ   r_   rj   r}   r   r   r3   r/   r  r    s5    0E.(#$0r3   r  r  z	A Poisson)r   rf  c                   X    \ rS rSrSrS rS rS rS rS r	S r
S	 rSS jrS rS rSrg
)
planck_geni  a  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                 @    [        SSS[        R                  4S5      /$ )Nlambda_Fr   r   r*   r-   s    r/   r0   planck_gen._shape_info(  s    9ea[.IJJr3   c                     US:  $ r   r   )r.   r  s     r/   r@   planck_gen._argcheck+  s    {r3   c                 <    [        U* 5      * [        U* U-  5      -  $ r5   )r   r   )r.   rM   r  s      r/   rU   planck_gen._pmf.  s     whWHQJ//r3   c                 >    [        U5      n[        U* US-   -  5      * $ rH   )r   r   r.   rL   r  rM   s       r/   rZ   planck_gen._cdf1  s#    !Hwh!n%%%r3   c                 6    [        U R                  X5      5      $ r5   )r   rX  )r.   rL   r  s      r/   r_   planck_gen._sf5  s    4;;q*++r3   c                 *    [        U5      nU* US-   -  $ rH   r   r  s       r/   rX  planck_gen._logsf8  s    !Hx1~r3   c                     [        SU-  [        U* 5      -  S-
  5      nUS-
  R                  " U R                  U5      6 nU R	                  XB5      n[
        R                  " XQ:  XC5      $ )N      r   )r   r   cliprE   rZ   r+   rG  )r.   ri   r  r   r  r]  s         r/   rj   planck_gen._ppf<  s^    DL5!9,Q./a 1 1' :<yy(xx	5//r3   Nc                 @    [        U* 5      * nUR                  XBS9S-
  $ )NrC  r   )r   rD  )r.   r  r8   r9   r'   s        r/   r:   planck_gen._rvsB  s)    G8_%%a%3c99r3   c                     S[        U5      -  n[        U* 5      [        U* 5      S-  -  nS[        US-  5      -  nSS[        U5      -  -   nX#XE4$ )Nr   r   ro   r8  )r   r   r   )r.   r  rt   ru   rv   rw   s         r/   r}   planck_gen._statsG  sZ    uW~7(mUG8_q00tGCK  qgr3   c                 X    [        U* 5      * nU[        U* 5      -  U-  [        U5      -
  $ r5   )r   r   r   )r.   r  Cs      r/   r   planck_gen._entropyN  s/    G8_sG8}$Q&Q//r3   r   rm   )r   r   r   r   r   r0   r@   rU   rZ   r_   rX  rj   r:   r}   r   r   r   r3   r/   r  r    s:    6K0&,0:
0r3   r  planckzA discrete exponential c                   B    \ rS rSrSrS rS rS rS rS r	S r
S	 rS
rg)boltzmann_geniV  ak  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    [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ )Nr  Fr   r   rk  Tr*   r-   s    r/   r0   boltzmann_gen._shape_infol  s:    9ea[.I3q"&&k>BD 	Dr3   c                 0    US:  US:  -  [        U5      -  $ r   r>   r.   r  rk  s      r/   r@   boltzmann_gen._argcheckp  s    !A&Q77r3   c                 $    U R                   US-
  4$ rH   rC   r#  s      r/   rE   boltzmann_gen._get_supports  s    vvq1u}r3   c                 j    S[        U* 5      -
  S[        U* U-  5      -
  -  nU[        U* U-  5      -  $ rH   r   )r.   rM   r  rk  facts        r/   rU   boltzmann_gen._pmfv  s<     #wh-!C
O"34C
O##r3   c                 h    [        U5      nS[        U* US-   -  5      -
  S[        U* U-  5      -
  -  $ rH   )r   r   )r.   rL   r  rk  rM   s        r/   rZ   boltzmann_gen._cdf|  s9    !H#wh!n%%#whqj/(9::r3   c                    US[        U* U-  5      -
  -  n[        SU-  [        SU-
  5      -  S-
  5      nUS-
  R                  S[        R
                  5      nU R                  XbU5      n[        R                  " Xq:  Xe5      $ )Nr   r  r  )r   r   r   r  r+   r,   rZ   rG  )r.   ri   r  rk  qnewr   r  r]  s           r/   rj   boltzmann_gen._ppf  sv    !C
O#$DL3qv;.q01ac266*yy+xx	5//r3   c                    [        U* 5      n[        U* U-  5      nUSU-
  -  X$-  SU-
  -  -
  nUSU-
  S-  -  X"-  U-  SU-
  S-  -  -
  nSU-
  SU-
  -  nX7S-  -  X"-  U-  -
  nUSU-   -  US-  -  US-  U-  SU-   -  -
  n	XS-  -  n	USSU-  -   X3-  -   -  US-  -  US-  U-  SSU-  -   XD-  -   -  -
  n
X-  U-  n
XVX4$ )Nr   r   r   r   r  r8  r(  )r.   r  rk  zzNrt   ru   trmtrm2rv   rw   s              r/   r}   boltzmann_gen._stats  s   M'!_AYqtQrT{"Q
lQSVQrTAI--taclq&13r6!!WS!V^ad2gqtn,+!A#ac	]36!AqD2Iq2vbe|$<<Yr3   r   N)r   r   r   r   r   r0   r@   rE   rU   rZ   rj   r}   r   r   r3   r/   r  r  V  s+    *D8$;0r3   r  	boltzmannz!A truncated discrete exponential )r   rD   rf  c                   R    \ rS rSrSrS rS rS rS rS r	S r
S	 rSS jrS rSrg
)randint_geni  a+  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                     [        SS[        R                  * [        R                  4S5      [        SS[        R                  * [        R                  4S5      /$ )NlowTr   highr*   r-   s    r/   r0   randint_gen._shape_info  sH    5$"&&"&&(9>J64266'266):NKM 	Mr3   c                 :    X!:  [        U5      -  [        U5      -  $ r5   r>   r.   r:  r;  s      r/   r@   randint_gen._argcheck  s    
k#..T1BBBr3   c                     XS-
  4$ rH   r   r>  s      r/   rE   randint_gen._get_support  s    F{r3   c                     [         R                  " U5      [         R                  " U[         R                  S9U-
  -  n[         R                  " X:  X:  -  US5      $ )Nr   r  )r+   	ones_liker  int64rG  )r.   rM   r:  r;  r'   s        r/   rU   randint_gen._pmf  sD    LLOrzz$bhh?#EFxxah/B77r3   c                 0    [        U5      nXB-
  S-   X2-
  -  $ r$  r  )r.   rL   r:  r;  rM   s        r/   rZ   randint_gen._cdf  s    !H",,r3   c                     [        XU-
  -  U-   5      S-
  nUS-
  R                  X#5      nU R                  XRU5      n[        R                  " Xa:  XT5      $ rH   )r   r  rZ   r+   rG  )r.   ri   r:  r;  r   r  r]  s          r/   rj   randint_gen._ppf  sR    A$s*+a/*yyT*xx	5//r3   c                     [         R                  " U5      [         R                  " U5      pCX4-   S-
  S-  nX4-
  nXf-  S-
  S-  nSnSXf-  S-   -  Xf-  S-
  -  n	XWX4$ )Nr   r   r   g      (@r  g333333)r+   r  )
r.   r:  r;  m2m1rt   dru   rv   rw   s
             r/   r}   randint_gen._stats  sk    D!2::c?Bgmq GsQw$s#qsSy1r3   Nc                    [         R                  " U5      R                  S:X  a.  [         R                  " U5      R                  S:X  a
  [        XAX#S9$ Ub,  [         R                  " X5      n[         R                  " X#5      n[         R
                  " [        [        U5      [         R                  " [        5      /S9nU" X5      $ )z=An array of *size* random integers >= ``low`` and < ``high``.r   rC  )otypes)	r+   r  r8   r	   broadcast_to	vectorizer   r   r  )r.   r:  r;  r8   r9   randints         r/   r:   randint_gen._rvs  s    ::c?1$D)9)>)>!)C4CC
 //#,C??4.D,,w|\B')xx}o7s!!r3   c                     [        X!-
  5      $ r5   )r   r>  s      r/   r   randint_gen._entropy  s    4:r3   r   rm   )r   r   r   r   r   r0   r@   rE   rU   rZ   rj   r}   r:   r   r   r   r3   r/   r8  r8    s7    =~MC8
-0""r3   r8  rT  z#A discrete uniform (random integer)c                   :    \ rS rSrSrS rS
S jrS rS rS r	S	r
g)zipf_geni  aA  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                 @    [        SSS[        R                  4S5      /$ )NrD   Fr   r   r*   r-   s    r/   r0   zipf_gen._shape_infoA      3266{NCDDr3   Nc                      UR                  XS9$ ro  )zipf)r.   rD   r8   r9   s       r/   r:   zipf_gen._rvsD  s       ..r3   c                     US:  $ rH   r   r.   rD   s     r/   r@   zipf_gen._argcheckG  s    1ur3   c                     UR                  [        R                  5      nS[        R                  " US5      -  X* -  -  nU$ Nr   r   )r   r+   float64r   zeta)r.   rM   rD   r  s       r/   rU   zipf_gen._pmfJ  s7    HHRZZ 7<<1%%2-	r3   c                 Z    [         R                  " X!S-   :  X!4S [        R                  S9$ )Nr   c                 d    [         R                  " X-
  S5      [         R                  " U S5      -  $ rH   )r   rf  )rD   r%   s     r/   r   zipf_gen._munp.<locals>.<lambda>S  s!    aeQ/',,q!2DDr3   r(  r9  )r.   r%   rD   s      r/   _munpzipf_gen._munpP  s*    AIvDvv 	r3   r   rm   )r   r   r   r   r   r0   r:   r@   rU   rk  r   r   r3   r/   rY  rY    s"    )VE/r3   rY  r^  zA Zipfc                   B    \ rS rSrSrS rS rS rS rS r	S r
S	 rS
rg)zipfian_geniZ  a  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 SciPy implementation of this distribution requires :math:`1 \le n \le 2^{53}`.
For larger values of :math:`n`, the `zipfian` methods (`pmf`, `cdf`, `mean`, etc.)
will return `nan`.

When :math:`a > 1`, 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    [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ )NrD   Fr   r&   r%   Tr   r*   r-   s    r/   r0   zipfian_gen._shape_info  s:    3266{MB3q"&&k>BD 	Dr3   c           	          US:  U[         R                  " [         R                  " USS5      5      R                  [         R                  S9:H  -  $ )Nr   r   l          rC  )r+   r  r  r   rE  r.   rD   r%   s      r/   r@   zipfian_gen._argcheck  sG     abjjAu!56==BHH=MMO 	Pr3   c                 2    S[         R                  " U5      4$ rH   )r+   r   rr  s      r/   rE   zipfian_gen._get_support  s    "((1+~r3   c                     [         R                  " U5      n[         R                  " U5      n[        R                  " XX25      $ r5   r+   r   rR   _normalized_gen_harmonicr.   rM   rD   r%   s       r/   rU   zipfian_gen._pmf  s/    HHQKHHQK++A!77r3   c                     [         R                  " U5      n[         R                  " U5      n[        R                  " SXU5      $ rH   rw  ry  s       r/   rZ   zipfian_gen._cdf  s1    HHQKHHQK++AqQ77r3   c                     [         R                  " U5      n[         R                  " U5      n[        R                  " US-   X3U5      $ rH   rw  ry  s       r/   r_   zipfian_gen._sf  s5    HHQKHHQK++AE1;;r3   c                    [         R                  " U5      n[        R                  " X!5      n[        R                  " X!S-
  5      n[        R                  " X!S-
  5      n[        R                  " X!S-
  5      n[        R                  " X!S-
  5      nXC-  nXS-  US-  -
  n	US-  n
X-  nXc-  SU-  U-  US-  -  -
  SUS-  -  US-  -  -   US-  -  nUS-  U-  SUS-  -  U-  U-  -
  SU-  US-  -  U-  -   SUS-  -  -
  U	S-  -  nUS-  nXX4$ )Nr   r   r   r8  r  r   )r+   r   rR   _gen_harmonic)r.   rD   r%   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2rv   rw   s                 r/   r}   zipfian_gen._stats  s;   HHQK%  aC(  aC(  aC(  aC(h47"Avkh4S!V++aaiQ.>>c
J1fTkAc1fHTM$..3tQwt1CC$'	!1W%
ar3   r   N)r   r   r   r   r   r0   r@   rE   rU   rZ   r_   r}   r   r   r3   r/   rn  rn  Z  s-    0dDP 8
8
<
 r3   rn  zipfianz	A Zipfianc                   F    \ rS rSrSrS rS rS rS rS r	S r
SS
 jrSrg	)dlaplace_geni  a   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                 @    [        SSS[        R                  4S5      /$ )NrD   Fr   r   r*   r-   s    r/   r0   dlaplace_gen._shape_info  r\  r3   c                 P    [        US-  5      [        U* [        U5      -  5      -  $ Nro   )r   r   abs)r.   rM   rD   s      r/   rU   dlaplace_gen._pmf  s$    AcE{S!c!f---r3   c                 \    [        U5      nS nS n[        R                  " US:  X24XE5      $ )Nc                 D    S[        U* U -  5      [        U5      S-   -  -
  $ rd  r(  rM   rD   s     r/   r   dlaplace_gen._cdf.<locals>.f1  s$    aR!VA
333r3   c                 @    [        XS-   -  5      [        U5      S-   -  $ rH   r(  r  s     r/   f2dlaplace_gen._cdf.<locals>.f2  s     qE{#s1vz22r3   r   )r   r:  r;  )r.   rL   rD   rM   r   r  s         r/   rZ   dlaplace_gen._cdf  s0    !H	4	3 qAvvr66r3   c           
      *   S[        U5      -   n[        [        R                  " USS[        U* 5      -   -  :  [	        X-  5      U-  S-
  [	        SU-
  U-  5      * U-  5      5      nUS-
  n[        R                  " U R                  XR5      U:  XT5      $ )Nr   r   )r   r   r+   rG  r   rZ   )r.   ri   rD   constr   r  s         r/   rj   dlaplace_gen._ppf  s    CF
BHHQCG!44 \A-1!1Q3%-00146 7 qxx		%+q0%>>r3   c                     [        U5      nSU-  US-
  S-  -  nSU-  US-  SU-  -   S-   -  US-
  S-  -  nSUSXCS-  -  S-
  4$ )Nro   r   r   g      $@r8  r  r-  r(  )r.   rD   ear  r  s        r/   r}   dlaplace_gen._stats  se    VeRUQJeRU3r6\"_%B
23CQJO++r3   c                 N    U[        U5      -  [        [        US-  5      5      -
  $ r  )r   r   r   ra  s     r/   r   dlaplace_gen._entropy   s"    47{Sae---r3   Nc                     [         R                  " [         R                  " U5      * 5      * nUR                  XBS9nUR                  XBS9nXV-
  $ ro  )r+   r   r  rD  )r.   rD   r8   r9   probOfSuccessrL   ys          r/   r:   dlaplace_gen._rvs  sL      2::a=.11""="<""="<ur3   r   rm   )r   r   r   r   r   r0   rU   rZ   rj   r}   r   r:   r   r   r3   r/   r  r    s+    ,E.	7?,.r3   r  dlaplacezA discrete Laplacianc                   R    \ rS rSrSrS rS rSSS.S jrS rS	 r	S
 r
S rS rSrg)poisson_binom_geni  u  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                     / $ r5   r   r-   s    r/   r0   poisson_binom_gen._shape_infoF  s	     	r3   c                 l    [         R                  " USS9nSU:*  US:*  -  n[         R                  " USS9$ Nr   r   r   )r+   stackall)r.   argsr'   condss       r/   r@   poisson_binom_gen._argcheckK  s5    HHT"aAF#vve!$$r3   Nr   c                (   [         R                  " USS9nUc  UR                  O,[         R                  " U5      (       a  US4O[	        U5      S-   n[         R
                  " UR                  U5      n[        R                  XAUS9R                  SS9$ )Nr   r   )r   r   )	r+   r  shapeisscalartuplebroadcast_shapesr   r:   r   )r.   r8   r9   r  r'   s        r/   r:   poisson_binom_gen._rvsP  s|    HHT#  <[[..q	E$K$4F 	""177D1~~a~FJJPRJSSr3   c                     S[        U5      4$ r   )len)r.   r  s     r/   rE   poisson_binom_gen._get_supportZ  s    #d)|r3   c                     [         R                  " U5      R                  [         R                  5      n[         R                  " U/UQ76 tp[         R
                  " U[         R                  S9n[        XS5      $ )NrC  r  r+   
atleast_1dr   rE  r   r  re  r   r.   rM   r  s      r/   rU   poisson_binom_gen._pmf]  W    MM!##BHH-&&q040zz$bjj1au--r3   c                     [         R                  " U5      R                  [         R                  5      n[         R                  " U/UQ76 tp[         R
                  " U[         R                  S9n[        XS5      $ )NrC  r   r  r  s      r/   rZ   poisson_binom_gen._cdfc  r  r3   c                     [         R                  " USS9n[         R                  " USS9n[         R                  " USU-
  -  SS9nXES S 4$ r  )r+   r  r   )r.   r  kwdsr'   r&  ru   s         r/   r}   poisson_binom_gen._statsi  sG    HHT"vvaa ffQ!A#YQ'4&&r3   c                      [        U /UQ70 UD6$ r5   )poisson_binomial_frozen)r.   r  r  s      r/   __call__poisson_binom_gen.__call__o  s    &t;d;d;;r3   r   )r   r   r   r   r   r0   r@   r:   rE   rU   rZ   r}   r  r   r   r3   r/   r  r    s8    'P
%
  $$ T..'<r3   r  poisson_binomzA Poisson binomialr'   )r   rf  shapesc                 L    [        [        R                  " USS5      5      USU4$ Nr  r   r   r  r+   moveaxis)r.   r'   locr8   s       r/   _parse_args_rvsr  {  s#    QA&'c477r3   c                 L    [        [        R                  " USS5      5      USU4$ r  r  )r.   r'   r  rs   s       r/   _parse_args_statsr  ~  s#    QA&'c7::r3   c                 J    [        [        R                  " USS5      5      US4$ r  r  )r.   r'   r  s      r/   _parse_argsr    s!    QA&'c11r3   c                   $    \ rS rSrS rSS jrSrg)r  i  c                     X l         X0l        UR                  " S0 UR                  5       D6U l        [
        R                  [        [        5      U R                  l        [        R                  [        [        5      U R                  l	        [        R                  [        [        5      U R                  l
        U R                  R                  " U0 UD6u  n  nU R                  R                  " U6 u  U l        U l        g )Nr   )r  r  	__class___updated_ctor_paramdistr  __get___pb_obj_pb_clsr  r  rE   rD   r   )r.   r  r  r  r  _s         r/   __init__ poisson_binomial_frozen.__init__  s    		 NN@T%=%=%?@	 %4$;$;GW$M		!&7&?&?&Q		# + 3 3GW E		yy,,d;d;1//8r3   Nc                     U R                   R                  " U R                  0 U R                  D6u  pgnU R                   R                  " XR                  XrX440 UD6$ r5   )r  r  r  r  expect)	r.   funclbubconditionalr  rD   r  scales	            r/   r  poisson_binomial_frozen.expect  sK    		--tyyFDIIF yyii"RTRRr3   )rD   r  r   r  r  )NNNF)r   r   r   r   r  r  r   r   r3   r/   r  r    s    9Sr3   r  c                   :    \ rS rSrSrS rS
S jrS rS rS r	S	r
g)skellam_geni  a  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    [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ )Nr  Fr   r   r  r*   r-   s    r/   r0   skellam_gen._shape_info  s:    5%!RVVnE5%!RVVnEG 	Gr3   Nc                 L    UnUR                  X5      UR                  X%5      -
  $ r5   r  )r.   r  r  r8   r9   r%   s         r/   r:   skellam_gen._rvs  s-    $$S,$$S,- 	.r3   c                 .   [         R                  " SS9   [         R                  " US:  [        R                  " SU-  SSU-
  -  SU-  5      S-  [        R                  " SU-  SSU-   -  SU-  5      S-  5      nS S S 5        U$ ! , (       d  f       W$ = f)Nr   r  r   r   r   )r+   r   rG  rR   	_ncx2_pdfr.   rL   r  r  pxs        r/   rU   skellam_gen._pmf  s    [[h'!a%--#q!A#w#>q@--#q!A#w#>q@BB (
 	 ('
 	s   A&B
Bc                 2   [        U5      n[        R                  " SS9   [        R                  " US:  [        R
                  " SU-  SU-  SU-  5      [        R                  " SU-  SUS-   -  SU-  5      5      nS S S 5        U$ ! , (       d  f       W$ = f)Nr   r  r   r   r   )r   r+   r   rG  r   chndtrrR   _ncx2_sfr  s        r/   rZ   skellam_gen._cdf  s    !H[[h'!a%!..31ae<,,qua1gqu=?B ( 		 (' 	s   AB
Bc                 F    X-
  nX-   nU[        US-  5      -  nSU-  nX4XV4$ )Nr   r   r.  )r.   r  r  r&  ru   rv   rw   s          r/   r}   skellam_gen._stats  s6    yiD#N"W"  r3   r   rm   )r   r   r   r   r   r0   r:   rU   rZ   r}   r   r   r3   r/   r  r    s!    :G.
!r3   r  skellamz	A Skellamc                   R    \ rS rSrSrS rSS jrS rS rS r	S	 r
S
 rS rS rSrg)yulesimon_geni  a  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                 @    [        SSS[        R                  4S5      /$ )NalphaFr   r   r*   r-   s    r/   r0   yulesimon_gen._shape_info  s    7EArvv;GHHr3   Nc           	          UR                  U5      nUR                  U5      n[        U* [        [        U* U-  5      * 5      -  5      nU$ r5   )standard_exponentialr   r   r   )r.   r  r8   r9   E1E2anss          r/   r:   yulesimon_gen._rvs
  sK    ..t4..t4B3RC%K 00112
r3   c                 :    U[         R                  " XS-   5      -  $ rH   r   r   r.   rL   r  s      r/   rU   yulesimon_gen._pmf  s    w||Aqy111r3   c                     US:  $ r   r   )r.   r  s     r/   r@   yulesimon_gen._argcheck  s    	r3   c                 L    [        U5      [        R                  " XS-   5      -   $ rH   r   r   r   r  s      r/   rO   yulesimon_gen._logpmf  s    5zGNN1ai888r3   c                 @    SU[         R                  " XS-   5      -  -
  $ rH   r  r  s      r/   rZ   yulesimon_gen._cdf  s    1w||Aqy1111r3   c                 :    U[         R                  " XS-   5      -  $ rH   r  r  s      r/   r_   yulesimon_gen._sf  s    7<<19---r3   c                 L    [        U5      [        R                  " XS-   5      -   $ rH   r  r  s      r/   rX  yulesimon_gen._logsf  s    1vq!)444r3   c                    [         R                  " US:*  [         R                  XS-
  -  5      n[         R                  " US:  US-  US-
  US-
  S-  -  -  [         R                  5      n[         R                  " US:*  [         R                  U5      n[         R                  " US:  [	        US-
  5      US-   S-  -  XS-
  -  -  [         R                  5      n[         R                  " US:*  [         R                  U5      n[         R                  " US:  US-   SUS-  -  SU-  -
  S-
  XS-
  -  US-
  -  -  -   [         R                  5      n[         R                  " US:*  [         R                  U5      nX#XE4$ )	Nr   r   ro   r   r8     1      )r+   rG  r,   nanr   )r.   r  rt   r  rv   rw   s         r/   r}   yulesimon_gen._stats"  sQ   XXeqj"&&%19*=>hhuqyaxECKEAI>#ABvv hhuz2663/XXeai519oQ6%19:MNff XXeqj"&&"-XXeaiaiBMBJ$>$C$)QY$7519$E$G Hff XXeqj"&&"-r3   r   rm   )r   r   r   r   r   r0   r:   rU   r@   rO   rZ   r_   rX  r}   r   r   r3   r/   r   r     s6     BI292.5r3   r   	yulesimon)r   rD   c                   L    \ rS rSrSrSrSrS rS rS r	SS jr
S rSS	 jrS
rg)_nchypergeom_geni7  zA noncentral hypergeometric discrete random variable.

For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.

Nc           	          [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S	5      /$ )
Nrj  Tr   r&   r%   rk  oddsFr   r*   r-   s    r/   r0   _nchypergeom_gen._shape_infoA  sf    3q"&&k=A3q"&&k=A3q"&&k=A651bff+~FH 	Hr3   c                 v    XUp%nX5-
  n[         R                  " SX&-
  5      n[         R                  " X%5      nXx4$ r   rs  )	r.   rj  r%   rk  r#  rM  rL  x_minx_maxs	            r/   rE   _nchypergeom_gen._get_supportG  s:    qV

1af%

1!|r3   c                 ,   [         R                  " U5      [         R                  " U5      p![         R                  " U5      [         R                  " U5      pC[         R                  " U5      ) UR                  [        5      U:H  -  US:  -  n[         R                  " U5      ) UR                  [        5      U:H  -  US:  -  n[         R                  " U5      ) UR                  [        5      U:H  -  US:  -  nUS:  nX1:*  n	X!:*  n
XV-  U-  U-  U	-  U
-  $ r   )r+   r  isnanr   r  )r.   rj  r%   rk  r#  cond1cond2cond3cond4cond5cond6s              r/   r@   _nchypergeom_gen._argcheckN  s    zz!}bjjm1**Q-D!14((1+!((3-1"45a@((1+!((3-1"45a@((1+!((3-1"45a@q}u$u,u4u<<r3   c           	      2   ^  [         U 4S j5       nU" XX4XVS9$ )Nc                   > [         R                  " U 5      [         R                  " U5      -  [         R                  " U5      -  (       a%  [         R                  " U[         R                  5      $ [         R                  " U5      n[        5       n[        UT
R                  5      nU" X!XXe5      n	U	R                  U5      n	U	$ r5   )	r+   r*  fullr  prodr   getattrrvs_namereshape)rj  r%   rk  r#  r8   r9   lengthurnrv_genr  r.   s             r/   r  $_nchypergeom_gen._rvs.<locals>._rvs1[  s    xx{RXXa[(288A;6wwtRVV,,WWT]F#%CS$--0Fq=C++d#CJr3   r   r  )r.   rj  r%   rk  r#  r8   r9   r  s   `       r/   r:   _nchypergeom_gen._rvsY  s&    	#	 
$	 Q1IIr3   c                    ^  [         R                  " XX4U5      u  pp4nUR                  S:X  a  [         R                  " U5      $ [         R                  U 4S j5       nU" XX4U5      $ )Nr   c                 ,  > [         R                  " U 5      [         R                  " U5      -  [         R                  " U5      -  [         R                  " U5      -  (       a  [         R                  $ TR                  X2XS5      nUR	                  U 5      $ Ng-q=)r+   r*  r  r  probability)rL   rj  r%   rk  r#  r:  r.   s         r/   _pmf1$_nchypergeom_gen._pmf.<locals>._pmf1n  s_    xx{RXXa[(288A;6!Dvv))A!51C??1%%r3   )r+   r   r8   
empty_likerS  )r.   rL   rj  r%   rk  r#  rB  s   `      r/   rU   _nchypergeom_gen._pmfh  s_    ..qQ4@aD66Q;==##		& 
	& Q1&&r3   c                 x   ^  [         R                  U 4S j5       nSU;   d  SU;   a	  U" XX45      OSu  pxSu  pXxX4$ )Nc                   > [         R                  " U 5      [         R                  " U5      -  [         R                  " U5      -  (       a   [         R                  [         R                  4$ TR                  X!XS5      nUR	                  5       $ r@  )r+   r*  r  r  rs   )rj  r%   rk  r#  r:  r.   s        r/   	_moments1*_nchypergeom_gen._stats.<locals>._moments1y  sX    xx{RXXa[(288A;6vvrvv~%))A!51C;;= r3   r  vrm   )r+   rS  )r.   rj  r%   rk  r#  rs   rH  r  rJ  rn   rM   s   `          r/   r}   _nchypergeom_gen._statsw  sL    		! 
	! .1G^sg~	!(! 	Qzr3   r   rm   r   )r   r   r   r   r   r7  r  r0   rE   r@   r:   rU   r}   r   r   r3   r/   r!  r!  7  s3     HDH	=J'r3   r!  c                        \ rS rSrSrSr\rSrg)nchypergeom_fisher_geni  a  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_fisherr   N)	r   r   r   r   r   r7  r   r  r   r   r3   r/   rM  rM    s    GR H%Dr3   rM  nchypergeom_fisherz$A Fisher's noncentral hypergeometricc                        \ rS rSrSrSr\rSrg)nchypergeom_wallenius_geni  a  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_walleniusr   N)	r   r   r   r   r   r7  r   r  r   r   r3   r/   rQ  rQ    s    GR H'Dr3   rQ  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)r   N)r   r   )r   )i	functoolsr   scipyr   scipy.specialr   r   r   r   rI   scipy.special._ufuncs_ufuncsrR   scipy._lib._utilr	   scipy._lib.array_api_extra_libarray_api_extrar:  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   r!   r   r   r   r   r   r   r  r  r=  r?  re  rh  r  r  r  r  r  r  r  r  r  r  r6  r8  rT  rY  r^  rn  r  r  r,   r  r  r  r  r  r  r  r  r  r  r  r  r   r  r!  rM  rO  rQ  rS  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r   r3   r/   <module>rj     s  
   C C # # ) ( ( & M M M 8 8, , +]* ]*@ 	w>#I >#B AK0	OK Od {+	r
 r
j 
	"Z[ Zz .
N7{ N7b !&=9XK Xv {+	[[ [| .
= =@ 
ah	A?+ ?D 9{
;D0 D0N 
ah1J	K<K <~ {a#FH	t+ tn 9 0) *
?{ ?D !&84i + i X 	K
@M; M` 266''2HJS< S<l "AU),.8;2
 !"3  / 7 7 I "3";";GW"M '//A S0 S0<!+ <!~ i+
FLK L^ {a0	L{ L^K&- K&\ ,	35 
K( 0 K(\ 2	 57  	WY^^##%&!7{!K 
)
)r3   