
    (phG                     h   S SK Jr  S SKJr  S SKJrJrJrJr	J
r
  S SKJr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K)Js  J*r+   " S S\5      r,\," SS9r- " S S\,5      r.\." SSS9r/ " S S\5      r0\0" 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S S!9r7 " S" S#\5      r8\8" S$S9r9 " 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r? " S1 S2\5      r@\@" S S3S4S!9rA " S5 S6\5      rB\B" S7S S8S99rC " S: S;\5      rD\D" S<S=S09rE " S> S?\5      rF\F" SS@SAS!9rGSB rHSC rISD rJ " SE SF\5      rK\K" SSGSHS!9rL " SI SJ\5      rM\M" \R                  * SKSLS!9rO " SM SN\5      rP\P" SOSPSQSR9rQSjSS jrRSkST jrSSlSU jrT\Q\PsrUrV\RR                  \U\V5      \QlR        \SR                  \U\V5      \QlS        \TR                  \U\V5      \QlT         " SV SW\"5      rX " SX SY\5      rY\Y" \R                  * SZS[S!9rZ " S\ S]\5      r[\[" S^SS_9r\ " S` Sa\5      r] " Sb Sc\]5      r^\^" SdSeS09r_ " Sf Sg\]5      r`\`" ShSiS09ra\b" \c" 5       R                  5       R                  5       5      rf\" \f\5      u  rgrh\g\h-   rig)m    )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                   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      /$ 	NnTr   TFpFr   r   TTr   npinfselfs    O/var/www/html/venv/lib/python3.13/site-packages/scipy/stats/_discrete_distns.py_shape_infobinom_gen._shape_infoA   0    3q"&&k=A3v|<> 	>    Nc                 &    UR                  XU5      $ N)binomialr0   r'   r)   sizerandom_states        r1   _rvsbinom_gen._rvsE   s    $$Q400r5   c                 <    US:  [        U5      -  US:  -  US:*  -  $ Nr   r   r   r0   r'   r)   s      r1   	_argcheckbinom_gen._argcheckH   s'    Q+a.(AF3qAv>>r5   c                     U R                   U4$ r7   arA   s      r1   _get_supportbinom_gen._get_supportK   s    vvqyr5   c                     [        U5      n[        US-   5      [        US-   5      [        X$-
  S-   5      -   -
  nU[        R                  " XC5      -   [        R                  " X$-
  U* 5      -   $ Nr   )r   gamlnr   xlogyxlog1py)r0   xr'   r)   kcombilns         r1   _logpmfbinom_gen._logpmfN   s\    !H1:qseACEl!:;q,,wqsQB/GGGr5   c                 0    [         R                  " XU5      $ r7   )scu
_binom_pmfr0   rN   r'   r)   s       r1   _pmfbinom_gen._pmfS   s    ~~aA&&r5   c                 F    [        U5      n[        R                  " XBU5      $ r7   )r   rT   
_binom_cdfr0   rN   r'   r)   rO   s        r1   _cdfbinom_gen._cdfW       !H~~aA&&r5   c                 F    [        U5      n[        R                  " XBU5      $ r7   )r   rT   	_binom_sfr[   s        r1   _sfbinom_gen._sf[   s    !H}}Q1%%r5   c                 0    [         R                  " XU5      $ r7   )rT   
_binom_isfrV   s       r1   _isfbinom_gen._isf_       ~~aA&&r5   c                 0    [         R                  " XU5      $ r7   )rT   
_binom_ppfr0   qr'   r)   s       r1   _ppfbinom_gen._ppfb   rg   r5   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$ )NNNs       @rO         @)r-   squarer   
reciprocal)r0   r'   r)   momentsmuvarg1g2pqnpq_sqrtt1t2npqs                r1   _statsbinom_gen._statse   s    Uryy|##'>RYYq\!BwwqvHx(B'X%BB'>RYYq\!B&Cs#BQBBr5   c                     [         R                  SUS-    nU R                  X1U5      n[         R                  " [	        U5      SS9$ )Nr   r   axis)r-   r_rW   sumr   )r0   r'   r)   rO   valss        r1   _entropybinom_gen._entropyw   s:    EE!AENyyq!vvd4jq))r5    ro   mv__name__
__module____qualname____firstlineno____doc__r2   r<   rB   rG   rQ   rW   r\   ra   re   rl   r   r   __static_attributes__r   r5   r1   r#   r#      sE    "F>1?H
''&''$*r5   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      /$ Nr)   Fr*   r+   r   r/   s    r1   r2   bernoulli_gen._shape_info       3v|<==r5   Nc                 .    [         R                  U SXUS9$ )Nr   r:   r;   )r#   r<   r0   r)   r:   r;   s       r1   r<   bernoulli_gen._rvs   s    ~~dAq,~OOr5   c                     US:  US:*  -  $ r?   r   r0   r)   s     r1   rB   bernoulli_gen._argcheck   s    Q16""r5   c                 2    U R                   U R                  4$ r7   )rF   br   s     r1   rG   bernoulli_gen._get_support   s    vvtvv~r5   c                 0    [         R                  USU5      $ rJ   )r   rQ   r0   rN   r)   s      r1   rQ   bernoulli_gen._logpmf   s    }}Q1%%r5   c                 0    [         R                  USU5      $ rJ   )r   rW   r   s      r1   rW   bernoulli_gen._pmf   s     zz!Q""r5   c                 0    [         R                  USU5      $ rJ   )r   r\   r   s      r1   r\   bernoulli_gen._cdf       zz!Q""r5   c                 0    [         R                  USU5      $ rJ   )r   ra   r   s      r1   ra   bernoulli_gen._sf   s    yyAq!!r5   c                 0    [         R                  USU5      $ rJ   )r   re   r   s      r1   re   bernoulli_gen._isf   r   r5   c                 0    [         R                  USU5      $ rJ   )r   rl   )r0   rk   r)   s      r1   rl   bernoulli_gen._ppf   r   r5   c                 .    [         R                  SU5      $ rJ   )r   r   r   s     r1   r   bernoulli_gen._stats   s    ||Aq!!r5   c                 6    [        U5      [        SU-
  5      -   $ rJ   )r   r   s     r1   r   bernoulli_gen._entropy   s    Awac""r5   r   ro   r   r   r5   r1   r   r      sD    0>P#&#
#"##"#r5   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.

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

%(after_notes)s

.. 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      /$ 	Nr'   Tr   r(   rF   FFFr   r,   r/   s    r1   r2   betabinom_gen._shape_info   P    3q"&&k=A3266{NC3266{NCE 	Er5   Nc                 J    UR                  X#U5      nUR                  XU5      $ r7   )betar8   r0   r'   rF   r   r:   r;   r)   s          r1   r<   betabinom_gen._rvs   s'    aD)$$Q400r5   c                 
    SU4$ Nr   r   r0   r'   rF   r   s       r1   rG   betabinom_gen._get_support       !tr5   c                 <    US:  [        U5      -  US:  -  US:  -  $ r   r@   r   s       r1   rB   betabinom_gen._argcheck   '    Q+a.(AE2a!e<<r5   c                     [        U5      n[        US-   5      * [        X%-
  S-   US-   5      -
  nU[        XS-   X%-
  U-   5      -   [        X45      -
  $ rJ   )r   r   r   r0   rN   r'   rF   r   rO   rP   s          r1   rQ   betabinom_gen._logpmf   sS    !Hq1u:+quqy!a% 88quqy11F1L@@r5   c                 8    [        U R                  XX45      5      $ r7   r   rQ   r0   rN   r'   rF   r   s        r1   rW   betabinom_gen._pmf       4<<a+,,r5   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$ )
Nr   ro   rp         ?   rO            )r   astypedtype)r0   r'   rF   r   ru   e_pe_qrv   rw   rx   ry   s              r1   r   betabinom_gen._stats   s   q5k#gWq519o#c)QUQY7'>tCyBq51q5=QU++Bq519''B'>%		*Bq519q1u$%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r5   r   ro   r   )r   r   r   r   r   r2   r<   rG   rB   rQ   rW   r   r   r   r5   r1   r   r      s-    "FE
1=A
-r5   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      /$ r&   r,   r/   s    r1   r2   nbinom_gen._shape_infoT  r4   r5   Nc                 &    UR                  XU5      $ r7   )negative_binomialr9   s        r1   r<   nbinom_gen._rvsX  s    --aD99r5   c                 $    US:  US:  -  US:*  -  $ r?   r   rA   s      r1   rB   nbinom_gen._argcheck[  s    A!a% AF++r5   c                 0    [         R                  " XU5      $ r7   )rT   _nbinom_pmfrV   s       r1   rW   nbinom_gen._pmf^  s    qQ''r5   c                     [        X!-   5      [        US-   5      -
  [        U5      -
  nXB[        U5      -  -   [        R                  " X* 5      -   $ rJ   )rK   r   r   rM   )r0   rN   r'   r)   coeffs        r1   rQ   nbinom_gen._logpmfb  sD    ac
U1Q3Z'%(2Qx'//!R"888r5   c                 F    [        U5      n[        R                  " XBU5      $ r7   )r   rT   _nbinom_cdfr[   s        r1   r\   nbinom_gen._cdff  s    !HqQ''r5   c                 B   [        U5      n[        R                  " XBU5      u  pBnU R                  XBU5      nUS:  nS nUn[        R                  " SS9   U" XF   X&   X6   5      X'   [        R
                  " XV)    5      X) '   S S S 5        U$ ! , (       d  f       U$ = f)N      ?c                 h    [         R                  " [        R                  " U S-   USU-
  5      * 5      $ rJ   )r-   r   r   betainc)rO   r'   r)   s      r1   f1nbinom_gen._logcdf.<locals>.f1o  s)    88W__QUAq1u==>>r5   ignore)divide)r   r-   broadcast_arraysr\   errstater   )	r0   rN   r'   r)   rO   cdfcondr   logcdfs	            r1   _logcdfnbinom_gen._logcdfj  s    !H%%aA.aiia Sy	? [[)agqw8FLFF3u:.F5M *  *) s   /B
Bc                 F    [        U5      n[        R                  " XBU5      $ r7   )r   rT   
_nbinom_sfr[   s        r1   ra   nbinom_gen._sfy  r^   r5   c                     [         R                  " SS9   [        R                  " XU5      sS S S 5        $ ! , (       d  f       g = fNr   over)r-   r   rT   _nbinom_isfrV   s       r1   re   nbinom_gen._isf}  (    [[h'??1+ (''	   6
Ac                     [         R                  " SS9   [        R                  " XU5      sS S S 5        $ ! , (       d  f       g = fr  )r-   r   rT   _nbinom_ppfrj   s       r1   rl   nbinom_gen._ppf  r  r  c                     [         R                  " X5      [         R                  " X5      [         R                  " X5      [         R                  " X5      4$ r7   )rT   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessrA   s      r1   r   nbinom_gen._stats  sD    Q"  &  &''-	
 	
r5   r   ro   )r   r   r   r   r   r2   r<   rB   rW   rQ   r\   r  ra   re   rl   r   r   r   r5   r1   r   r     s?    9t>:,(9(',,
r5   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.

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                     [        SSS[        R                  4S5      [        SSS[        R                  4S5      [        SSS[        R                  4S5      /$ r   r,   r/   s    r1   r2   betanbinom_gen._shape_info  r   r5   Nc                 J    UR                  X#U5      nUR                  XU5      $ r7   )r   r   r   s          r1   r<   betanbinom_gen._rvs  s'    aD)--aD99r5   c                 <    US:  [        U5      -  US:  -  US:  -  $ r   r@   r   s       r1   rB   betanbinom_gen._argcheck  r   r5   c                     [        U5      n[        R                  " X%-   5      * [        X%S-   5      -
  nU[        X2-   XE-   5      -   [        X45      -
  $ rJ   )r   r-   r   r   r   s          r1   rQ   betanbinom_gen._logpmf  sI    !H66!%=.6!U#33qu--q<<r5   c                 8    [        U R                  XX45      5      $ r7   r   r   s        r1   rW   betanbinom_gen._pmf  r   r5   c                 :   S n[        US:  XU4U[        R                  S9nS n[        US:  XU4U[        R                  S9nSu  pS n
SU;   a  [        US	:  XU4U
[        R                  S9nS
 nSU;   a  [        US:  XU4U[        R                  S9n	XgX4$ )Nc                     X-  US-
  -  $ Nr   r   r'   rF   r   s      r1   mean#betanbinom_gen._stats.<locals>.mean  s    5AF##r5   r   )f	fillvaluec                 H    X-  X-   S-
  -  X-   S-
  -  US-
  US-
  S-  -  -  $ )Nr   rq   r   r'  s      r1   rw   "betanbinom_gen._stats.<locals>.var  s;    EQURZ(AEBJ7B1r6B,.0 1r5   r   ro   c                     SU -  U-   S-
  SU-  U-   S-
  -  US-
  -  [        X-  X-   S-
  -  X!-   S-
  -  US-
  -  5      -  $ )Nr   r         @rq   r   r'  s      r1   skew#betanbinom_gen._stats.<locals>.skew  sf    UQY^A	B72v!%aequrz&:aebj&I2v' "   !r5   rp   r   c                 t   US-
  nUS-
  S-  US-  USU-  S-
  -  -   SUS-
  -  U-  -   -  SU S-  -  US-   US-  -  US-   US-
  -  U-  -   SUS-
  S-  -  -   -  -   SUS-
  -  U -  US-   US-  -  US-   US-
  -  U-  -   SUS-
  S-  -  -   -  -   nUS	-
  US-
  -  U-  U -  X-   S-
  -  X-   S-
  -  nX4-  U-  S-
  $ )
Nrq   r   r   rr   r/        @r   r   g      @r   )r'   rF   r   termterm_2denominators         r1   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3r5   rO      r
   r-   r.   )r0   r'   rF   r   ru   r(  rv   rw   rx   ry   r1  r8  s               r1   r   betanbinom_gen._stats  s    	$AayDBFFC	1 Qq	SBFFC	! '>AEA!9GB	4 '>AEA!9BFFKBr5   r   ro   r   )r   r   r   r   r   r2   r<   rB   rQ   rW   r   r   r   r5   r1   r  r    s'    #HE
:==
-!r5   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      /$ r   r   r/   s    r1   r2   geom_gen._shape_info  r   r5   Nc                     UR                  XS9n[        R                  " UR                  5      R                  n[        R
                  " US:  XT5      $ )Nr:   r   )	geometricr-   iinfor   maxwhere)r0   r)   r:   r;   resmax_ints         r1   r<   geom_gen._rvs  sD    $$Q$2 ((399%))xxa..r5   c                     US:*  US:  -  $ Nr   r   r   r   s     r1   rB   geom_gen._argcheck  s    Q1q5!!r5   c                 B    [         R                  " SU-
  US-
  5      U-  $ rJ   )r-   powerr0   rO   r)   s      r1   rW   geom_gen._pmf   s     xx!QqS!A%%r5   c                 P    [         R                  " US-
  U* 5      [        U5      -   $ rJ   )r   rM   r   rP  s      r1   rQ   geom_gen._logpmf#  s"    q1uqb)CF22r5   c                 J    [        U5      n[        [        U* 5      U-  5      * $ r7   )r   r   r   r0   rN   r)   rO   s       r1   r\   geom_gen._cdf&  s#    !HeQBik"""r5   c                 L    [         R                  " U R                  X5      5      $ r7   )r-   r   _logsfr   s      r1   ra   geom_gen._sf*  s    vvdkk!'((r5   c                 6    [        U5      nU[        U* 5      -  $ r7   )r   r   rU  s       r1   rX  geom_gen._logsf-  s    !Hr{r5   c                     [        [        U* 5      [        U* 5      -  5      nU R                  US-
  U5      n[        R                  " XA:  US:  -  US-
  U5      $ rL  )r   r   r\   r-   rG  )r0   rk   r)   r   temps        r1   rl   geom_gen._ppf1  sS    E1"Iqb	)*yya#xxtax0$q&$??r5   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   rq   )r   ir   )r   r-   polyval)r0   r)   rv   qrrw   rx   ry   s          r1   r   geom_gen._stats6  sT    UUfqj!etBxZZ
A&A.r5   c                 r    [         R                  " U5      * [         R                  " U* 5      SU-
  -  U-  -
  $ r&  )r-   r   r   r   s     r1   r   geom_gen._entropy>  s/    q	zBHHaRLCE2Q666r5   r   ro   )r   r   r   r   r   r2   r<   rB   rW   rQ   r\   ra   rX  rl   r   r   r   r   r5   r1   r?  r?    s@    B>/"&3#)@
7r5   r?  geomzA geometric)rF   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_geniE  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      /$ )NMTr   r(   r'   Nr,   r/   s    r1   r2   hypergeom_gen._shape_info  P    3q"&&k=A3q"&&k=A3q"&&k=AC 	Cr5   Nc                 (    UR                  X!U-
  X4S9$ NrC  )hypergeometric)r0   rj  r'   rk  r:   r;   s         r1   r<   hypergeom_gen._rvs  s    **1c1*@@r5   c                 f    [         R                  " X1U-
  -
  S5      [         R                  " X#5      4$ r   r-   maximumminimum)r0   rj  r'   rk  s       r1   rG   hypergeom_gen._get_support  s'    zz!qS'1%rzz!'777r5   c                     US:  US:  -  US:  -  nXBU:*  X1:*  -  -  nU[        U5      [        U5      -  [        U5      -  -  nU$ r   r@   )r0   rj  r'   rk  r   s        r1   rB   hypergeom_gen._argcheck  sU    A!q&!Q!V,aAF##AQ/+a.@@r5   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$ rJ   r   )	r0   rO   rj  r'   rk  totgoodbadresults	            r1   rQ   hypergeom_gen._logpmf  s    Tja#fSUA&66a19MM1dfQh'(*0Qa	*BCQ"# r5   c                 0    [         R                  " XXB5      $ r7   )rT   _hypergeom_pmfr0   rO   rj  r'   rk  s        r1   rW   hypergeom_gen._pmf      !!!--r5   c                 0    [         R                  " XXB5      $ r7   )rT   _hypergeom_cdfr  s        r1   r\   hypergeom_gen._cdf  r  r5   c                 n   SU-  SU-  SU-  p2nX-
  nXS-   -  SU-  X-
  -  -
  SU-  U-  -
  nXQS-
  U-  U-  -  nUSU-  U-  X-
  -  U-  SU-  S-
  -  -  nXRU-  X-
  -  U-  US-
  -  US-
  -  -  n[         R                  " X#U5      [         R                  " X#U5      [         R                  " X#U5      U4$ )Nr   r   rr   r4  r   rq   r/  )rT   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r0   rj  r'   rk  mry   s         r1   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!,	
 	
r5   c                     [         R                  X1U-
  -
  [        X#5      S-    nU R                  XAX#5      n[         R                  " [        U5      SS9$ )Nr   r   r   )r-   r   minpmfr   r   )r0   rj  r'   rk  rO   r   s         r1   r   hypergeom_gen._entropy  sE    EE!1u+c!i!m,xxa#vvd4jq))r5   c                 0    [         R                  " XXB5      $ r7   )rT   _hypergeom_sfr  s        r1   ra   hypergeom_gen._sf  s      q,,r5   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   rQ   asarrayr0   rO   rj  r'   rk  rH  quantr{  r|  drawk2s              r1   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#r5   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   rQ   r  r  s              r1   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#r5   r   ro   )r   r   r   r   r   r2   r<   rG   rB   rQ   rW   r\   r   r   ra   rX  r  r   r   r5   r1   rh  rh  E  sG    HRC
A8..
"*
-

r5   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      /$ )Nrj  Tr   r(   r'   rr,   r/   s    r1   r2   nhypergeom_gen._shape_infoH  rm  r5   c                 
    SU4$ r   r   )r0   rj  r'   r  s       r1   rG   nhypergeom_gen._get_supportM  r   r5   c                     US:  X!:*  -  US:  -  X1U-
  :*  -  nU[        U5      [        U5      -  [        U5      -  -  nU$ r   r@   )r0   rj  r'   r  r   s        r1   rB   nhypergeom_gen._argcheckP  sK    Q16"a1f-c:AQ/+a.@@r5   Nc                 2   ^  [         U 4S j5       nU" XX4US9$ )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)kind
fill_valuerC  )	supportr-   r  r   r   uniformr   intitem)rj  r'   r  r:   r;   rF   r   ksr   ppfrvsr0   s              r1   _rvs1"nhypergeom_gen._rvs.<locals>._rvs1W  s     <<a(DA1c"B((2!'C3MJCl***56==cBC|xxz!Jr5   r   r   )r0   rj  r'   r  r:   r;   r  s   `      r1   r<   nhypergeom_gen._rvsU  s&    	#		 
$		 Q1lCCr5   c                 <    US:H  US:H  -  n[        U) XX44S SS9nU$ )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      -
  $ rJ   rz  )rO   rj  r'   r  s       r1   <lambda>(nhypergeom_gen._logpmf.<locals>.<lambda>h  s    "(1a.6!#q>!A!'Aqs1uQw!7"8:@Qq!:L"M!'!QSU!3"46<QqS!n"Er5           )r+  )r
   )r0   rO   rj  r'   r  r   r~  s          r1   rQ   nhypergeom_gen._logpmfe  s:    aAF#TEA!<F '*+ r5   c                 8    [        U R                  XX45      5      $ r7   r   )r0   rO   rj  r'   r  s        r1   rW   nhypergeom_gen._pmfo  s     4<<a+,,r5   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$ )Nr   r   r   ro   r   )r0   rj  r'   r  rv   rw   rx   ry   s           r1   r   nhypergeom_gen._statst  sv     Q$1bdaSACE]1gaiACEACE?+q1!A;? r5   r   ro   )r   r   r   r   r   r2   rG   rB   r<   rQ   rW   r   r   r   r5   r1   r  r    s.    bHC

D -
r5   r  
nhypergeomc                   :    \ rS rSrSrS rS
S j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      /$ r   r   r/   s    r1   r2   logser_gen._shape_info  r   r5   Nc                      UR                  XS9$ ro  )	logseriesr   s       r1   r<   logser_gen._rvs  s     %%a%33r5   c                     US:  US:  -  $ r?   r   r   s     r1   rB   logser_gen._argcheck  s    A!a%  r5   c                 l    [         R                  " X!5      * S-  U-  [        R                  " U* 5      -  $ r&  )r-   rO  r   r   rP  s      r1   rW   logser_gen._pmf  s,    $q(7==!+<<<r5   c                    [         R                  " U* 5      nXS-
  -  U-  nU* U-  US-
  S-  -  nXCU-  -
  nU* U-  SU-   -  SU-
  S-  -  nUSU-  U-  -
  SUS-  -  -   nU[        R                  " US5      -  nU* U-  SUS-
  S-  -  SU-  US-
  S-  -  -
  SU-  U-  US-
  S-  -  -   -  n	U	SU-  U-  -
  SU-  U-  U-  -   SUS-  -  -
  n
XS-  -  S-
  nX5X4$ )	Nr   r   r         ?r   r   r:  r/  )r   r   r-   rO  )r0   r)   r  rv   mu2prw   mu3pmu3rx   mu4pmu4ry   s               r1   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r5   r   ro   )r   r   r   r   r   r2   r<   rB   rW   r   r   r   r5   r1   r  r    s     0>4
!=r5   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      /$ )Nrv   Fr   r(   r,   r/   s    r1   r2   poisson_gen._shape_info  s    4BFF]CDDr5   c                     US:  $ r   r   )r0   rv   s     r1   rB   poisson_gen._argcheck  s    Qwr5   Nc                 $    UR                  X5      $ r7   poisson)r0   rv   r:   r;   s       r1   r<   poisson_gen._rvs  s    ##B--r5   c                 V    [         R                  " X5      [        US-   5      -
  U-
  nU$ rJ   )r   rL   rK   )r0   rO   rv   Pks       r1   rQ   poisson_gen._logpmf  s'    ]]1!E!a%L025	r5   c                 6    [        U R                  X5      5      $ r7   r   )r0   rO   rv   s      r1   rW   poisson_gen._pmf  s    4<<&''r5   c                 D    [        U5      n[        R                  " X25      $ r7   )r   r   pdtrr0   rN   rv   rO   s       r1   r\   poisson_gen._cdf  s    !H||A""r5   c                 D    [        U5      n[        R                  " X25      $ r7   )r   r   pdtrcr  s       r1   ra   poisson_gen._sf  s    !H}}Q##r5   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  )r0   rk   rv   r   vals1r]  s         r1   rl   poisson_gen._ppf  sJ    GNN1)*

4!8Q'||E&xx	5//r5   c                     Un[         R                  " U5      nUS:  n[        XC4S [         R                  5      n[        XC4S [         R                  5      nXXV4$ )Nr   c                     [        SU -  5      $ r&  r0  rN   s    r1   r  $poisson_gen._stats.<locals>.<lambda>  s    d3q5kr5   c                     SU -  $ r&  r   r  s    r1   r  r    s    c!er5   )r-   r  r
   r.   )r0   rv   rw   tmp
mu_nonzerorx   ry   s          r1   r   poisson_gen._stats  sP    jjn1W

F,A266J
FORVVDr5   r   ro   )r   r   r   r   r   r2   rB   r<   rQ   rW   r\   ra   rl   r   r   r   r5   r1   r  r    s5    0E.(#$0r5   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      /$ )NlambdaFr   r   r,   r/   s    r1   r2   planck_gen._shape_info"  s    8UQKHIIr5   c                     US:  $ r   r   )r0   lambda_s     r1   rB   planck_gen._argcheck%  s    {r5   c                 <    [        U* 5      * [        U* U-  5      -  $ r7   )r   r   )r0   rO   r  s      r1   rW   planck_gen._pmf(  s     whWHQJ//r5   c                 >    [        U5      n[        U* US-   -  5      * $ rJ   )r   r   r0   rN   r  rO   s       r1   r\   planck_gen._cdf+  s#    !Hwh!n%%%r5   c                 6    [        U R                  X5      5      $ r7   )r   rX  )r0   rN   r  s      r1   ra   planck_gen._sf/  s    4;;q*++r5   c                 *    [        U5      nU* US-   -  $ rJ   r   r	  s       r1   rX  planck_gen._logsf2  s    !Hx1~r5   c                     [        SU-  [        U* 5      -  S-
  5      nUS-
  R                  " U R                  U5      6 nU R	                  XB5      n[
        R                  " XQ:  XC5      $ )N      r   )r   r   cliprG   r\   r-   rG  )r0   rk   r  r   r  r]  s         r1   rl   planck_gen._ppf6  s^    DL5!9,Q./a 1 1' :<yy(xx	5//r5   Nc                 @    [        U* 5      * nUR                  XBS9S-
  $ )NrC  r   )r   rD  )r0   r  r:   r;   r)   s        r1   r<   planck_gen._rvs<  s)    G8_%%a%3c99r5   c                     S[        U5      -  n[        U* 5      [        U* 5      S-  -  nS[        US-  5      -  nSS[        U5      -  -   nX#XE4$ )Nr   r   rq   r:  )r   r   r   )r0   r  rv   rw   rx   ry   s         r1   r   planck_gen._statsA  sZ    uW~7(mUG8_q00tGCK  qgr5   c                 X    [        U* 5      * nU[        U* 5      -  U-  [        U5      -
  $ r7   )r   r   r   )r0   r  Cs      r1   r   planck_gen._entropyH  s/    G8_sG8}$Q&Q//r5   r   ro   )r   r   r   r   r   r2   rB   rW   r\   ra   rX  rl   r<   r   r   r   r   r5   r1   r  r    s:    6J0&,0:
0r5   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_geniP  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      /$ )Nr  Fr   r   rk  Tr,   r/   s    r1   r2   boltzmann_gen._shape_infof  s:    9ea[.I3q"&&k>BD 	Dr5   c                 0    US:  US:  -  [        U5      -  $ r   r@   r0   r  rk  s      r1   rB   boltzmann_gen._argcheckj  s    !A&Q77r5   c                 $    U R                   US-
  4$ rJ   rE   r!  s      r1   rG   boltzmann_gen._get_supportm  s    vvq1u}r5   c                 j    S[        U* 5      -
  S[        U* U-  5      -
  -  nU[        U* U-  5      -  $ rJ   r   )r0   rO   r  rk  facts        r1   rW   boltzmann_gen._pmfp  s<     #wh-!C
O"34C
O##r5   c                 h    [        U5      nS[        U* US-   -  5      -
  S[        U* U-  5      -
  -  $ rJ   )r   r   )r0   rN   r  rk  rO   s        r1   r\   boltzmann_gen._cdfv  s9    !H#wh!n%%#whqj/(9::r5   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      $ )Nr   r  r  )r   r   r   r  r-   r.   r\   rG  )r0   rk   r  rk  qnewr   r  r]  s           r1   rl   boltzmann_gen._ppfz  sv    !C
O#$DL3qv;.q01ac266*yy+xx	5//r5   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  r:  r&  )r0   r  rk  zzNrv   rw   trmtrm2rx   ry   s              r1   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|$<<Yr5   r   N)r   r   r   r   r   r2   rB   rG   rW   r\   rl   r   r   r   r5   r1   r  r  P  s+    *D8$;0r5   r  	boltzmannz!A truncated discrete exponential )r   rF   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      /$ )NlowTr   highr,   r/   s    r1   r2   randint_gen._shape_info  sH    5$"&&"&&(9>J64266'266):NKM 	Mr5   c                 :    X!:  [        U5      -  [        U5      -  $ r7   r@   r0   r8  r9  s      r1   rB   randint_gen._argcheck  s    
k#..T1BBBr5   c                     XS-
  4$ rJ   r   r<  s      r1   rG   randint_gen._get_support  s    F{r5   c                     [         R                  " U5      [         R                  " U[         R                  S9U-
  -  n[         R                  " X:  X:  -  US5      $ )Nr   r  )r-   	ones_liker  int64rG  )r0   rO   r8  r9  r)   s        r1   rW   randint_gen._pmf  sD    LLOrzz$bhh?#EFxxah/B77r5   c                 0    [        U5      nXB-
  S-   X2-
  -  $ r&  r  )r0   rN   r8  r9  rO   s        r1   r\   randint_gen._cdf  s    !H",,r5   c                     [        XU-
  -  U-   5      S-
  nUS-
  R                  X#5      nU R                  XRU5      n[        R                  " Xa:  XT5      $ rJ   )r   r  r\   r-   rG  )r0   rk   r8  r9  r   r  r]  s          r1   rl   randint_gen._ppf  sR    A$s*+a/*yyT*xx	5//r5   c                     [         R                  " U5      [         R                  " U5      pCX4-   S-
  S-  nX4-
  nXf-  S-
  S-  nSnSXf-  S-   -  Xf-  S-
  -  n	XWX4$ )Nr   r   r   g      (@r  g333333)r-   r  )
r0   r8  r9  m2m1rv   drw   rx   ry   s
             r1   r   randint_gen._stats  sk    D!2::c?Bgmq GsQw$s#qsSy1r5   Nc                    [         R                  " U5      R                  S:X  a.  [         R                  " U5      R                  S:X  a
  [        XAX#S9$ 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  r:   r   broadcast_to	vectorizer   r   r  )r0   r8  r9  r:   r;   randints         r1   r<   randint_gen._rvs  s    ::c?1$D)9)>)>!)C4CC
 //#,C??4.D,,w|\B')xx}o7s!!r5   c                     [        X!-
  5      $ r7   )r   r<  s      r1   r   randint_gen._entropy  s    4:r5   r   ro   )r   r   r   r   r   r2   rB   rG   rW   r\   rl   r   r<   r   r   r   r5   r1   r6  r6    s7    =~MC8
-0""r5   r6  rR  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      /$ )NrF   Fr   r   r,   r/   s    r1   r2   zipf_gen._shape_info;      3266{NCDDr5   Nc                      UR                  XS9$ ro  )zipf)r0   rF   r:   r;   s       r1   r<   zipf_gen._rvs>  s       ..r5   c                     US:  $ rJ   r   r0   rF   s     r1   rB   zipf_gen._argcheckA  s    1ur5   c                     UR                  [        R                  5      nS[        R                  " US5      -  X* -  -  nU$ Nr   r   )r   r-   float64r   r	   )r0   rO   rF   r  s       r1   rW   zipf_gen._pmfD  s7    HHRZZ 7<<1%%2-	r5   c                 H    [        X!S-   :  X!4S [        R                  5      $ )Nr   c                 d    [         R                  " X-
  S5      [         R                  " U S5      -  $ rJ   )r   r	   )rF   r'   s     r1   r   zipf_gen._munp.<locals>.<lambda>M  s!    aeQ/',,q!2DDr5   r;  )r0   r'   rF   s      r1   _munpzipf_gen._munpJ  s&    AIvDFF 	r5   r   ro   )r   r   r   r   r   r2   r<   rB   rW   rh  r   r   r5   r1   rW  rW    s"    )VE/r5   rW  r\  zA Zipfc                 8    [        US5      [        XS-   5      -
  $ )z"Generalized harmonic number, a > 1r   )r	   r'   rF   s     r1   _gen_harmonic_gt1rl  T  s     1:Q!$$r5   c                    [         R                  " U 5      (       d  U $ [         R                  " U 5      n[         R                  " U[        S9n[         R
                  " USS[        S9 H  nX@:*  nX5==   SXAU   -  -  -  ss'   M     U$ )z#Generalized harmonic number, a <= 1rA  r   r   )r-   r:   rF  
zeros_likefloatr  )r'   rF   n_maxoutimasks         r1   _gen_harmonic_leq1ru  Z  sn    771::FF1IE
--
'CYYua51v	QqD'z\!	 2 Jr5   c                 b    [         R                  " X5      u  p[        US:  X4[        [        S9$ )zGeneralized harmonic numberr   r*  f2)r-   r   r
   rl  ru  rk  s     r1   _gen_harmonicry  g  s3    q$DAa!eaV).@B Br5   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)zipfian_genin  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 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      /$ )NrF   Fr   r(   r'   Tr   r,   r/   s    r1   r2   zipfian_gen._shape_info  s:    3266{MB3q"&&k>BD 	Dr5   c                 R    US:  US:  -  U[         R                  " U[        S9:H  -  $ )Nr   rA  )r-   r  r  r0   rF   r'   s      r1   rB   zipfian_gen._argcheck  s*    Q1q5!Q"**Qc*B%BCCr5   c                 
    SU4$ rJ   r   r  s      r1   rG   zipfian_gen._get_support  r   r5   c                 h    UR                  [        R                  5      nS[        X25      -  X* -  -  $ r&  )r   r-   rc  ry  r0   rO   rF   r'   s       r1   rW   zipfian_gen._pmf  s-    HHRZZ ]1((1b500r5   c                 0    [        X5      [        X25      -  $ r7   ry  r  s       r1   r\   zipfian_gen._cdf  s    Q"]1%888r5   c                 l    US-   nX-  [        X25      [        X5      -
  -  S-   X-  [        X25      -  -  $ rJ   r  r  s       r1   ra   zipfian_gen._sf  sB    E}Q*]1-@@AAE4a++- 	.r5   c                    [        X!5      n[        X!S-
  5      n[        X!S-
  5      n[        X!S-
  5      n[        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$ )Nr   r   r   r:  r  r   r  )r0   rF   r'   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2rx   ry   s                 r1   r   zipfian_gen._stats  s   A!Q!$Q!$Q!$Q!$h47"Avkh4S!V++aaiQ.>>c
J1fTkAc1fHTM$..3tQwt1CC$'	!1W%
ar5   r   N)r   r   r   r   r   r2   rB   rG   rW   r\   ra   r   r   r   r5   r1   r{  r{  n  s-    ,\DD19. r5   r{  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      /$ )NrF   Fr   r   r,   r/   s    r1   r2   dlaplace_gen._shape_info  rZ  r5   c                 P    [        US-  5      [        U* [        U5      -  5      -  $ Nrq   )r   r   abs)r0   rO   rF   s      r1   rW   dlaplace_gen._pmf  s$    AcE{S!c!f---r5   c                 B    [        U5      nS nS n[        US:  X24XES9$ )Nc                 D    S[        U* U -  5      [        U5      S-   -  -
  $ rb  r&  rO   rF   s     r1   r*  dlaplace_gen._cdf.<locals>.f  s$    aR!VA
333r5   c                 @    [        XS-   -  5      [        U5      S-   -  $ rJ   r&  r  s     r1   rx  dlaplace_gen._cdf.<locals>.f2  s     qE{#s1vz22r5   r   rw  )r   r
   )r0   rN   rF   rO   r*  rx  s         r1   r\   dlaplace_gen._cdf  s,    !H	4	3 !q&1&A55r5   c           
      *   S[        U5      -   n[        [        R                  " USS[        U* 5      -   -  :  [	        X-  5      U-  S-
  [	        SU-
  U-  5      * U-  5      5      nUS-
  n[        R                  " U R                  XR5      U:  XT5      $ )Nr   r   )r   r   r-   rG  r   r\   )r0   rk   rF   constr   r  s         r1   rl   dlaplace_gen._ppf  s    CF
BHHQCG!44 \A-1!1Q3%-00146 7 qxx		%+q0%>>r5   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$ )Nrq   r   r   g      $@r:  r  r/  r&  )r0   rF   ear  r  s        r1   r   dlaplace_gen._stats  se    VeRUQJeRU3r6\"_%B
23CQJO++r5   c                 N    U[        U5      -  [        [        US-  5      5      -
  $ r  )r   r   r   r_  s     r1   r   dlaplace_gen._entropy  s"    47{Sae---r5   Nc                     [         R                  " [         R                  " U5      * 5      * nUR                  XBS9nUR                  XBS9nXV-
  $ ro  )r-   r   r  rD  )r0   rF   r:   r;   probOfSuccessrN   ys          r1   r<   dlaplace_gen._rvs  sL      2::a=.11""="<""="<ur5   r   ro   )r   r   r   r   r   r2   rW   r\   rl   r   r   r<   r   r   r5   r1   r  r    s+    ,E.	6?,.r5   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                     / $ r7   r   r/   s    r1   r2   poisson_binom_gen._shape_infoG  s	     	r5   c                 l    [         R                  " USS9nSU:*  US:*  -  n[         R                  " USS9$ Nr   r   r   )r-   stackall)r0   argsr)   condss       r1   rB   poisson_binom_gen._argcheckL  s5    HHT"aAF#vve!$$r5   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$ )Nrn  r   r   )r   r   )	r-   r  shapeisscalartuplebroadcast_shapesr   r<   r   )r0   r:   r;   r  r)   s        r1   r<   poisson_binom_gen._rvsQ  s|    HHT#  <[[..q	E$K$4F 	""177D1~~a~FJJPRJSSr5   c                     S[        U5      4$ r   )len)r0   r  s     r1   rG   poisson_binom_gen._get_support[  s    #d)|r5   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      $ )NrA  r  r-   
atleast_1dr   rC  r   r  rc  r!   r0   rO   r  s      r1   rW   poisson_binom_gen._pmf^  W    MM!##BHH-&&q040zz$bjj1au--r5   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      $ )NrA  r   r  r  s      r1   r\   poisson_binom_gen._cdfd  r  r5   c                     [         R                  " USS9n[         R                  " USS9n[         R                  " USU-
  -  SS9nXES S 4$ r  )r-   r  r   )r0   r  kwdsr)   r(  rw   s         r1   r   poisson_binom_gen._statsj  sG    HHT"vvaa ffQ!A#YQ'4&&r5   c                      [        U /UQ70 UD6$ r7   )poisson_binomial_frozen)r0   r  r  s      r1   __call__poisson_binom_gen.__call__p  s    &t;d;d;;r5   r   )r   r   r   r   r   r2   rB   r<   rG   rW   r\   r   r  r   r   r5   r1   r  r    s8    'P
%
  $$ T..'<r5   r  poisson_binomzA Poisson binomialr)   )r   rf  shapesc                 L    [        [        R                  " USS5      5      USU4$ Nrn  r   r   r  r-   moveaxis)r0   r)   locr:   s       r1   _parse_args_rvsr  |  s#    QA&'c477r5   c                 L    [        [        R                  " USS5      5      USU4$ r  r  )r0   r)   r  ru   s       r1   _parse_args_statsr    s#    QA&'c7::r5   c                 J    [        [        R                  " USS5      5      US4$ r  r  )r0   r)   r  s      r1   _parse_argsr    s!    QA&'c11r5   c                   $    \ 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  rG   rF   r   )r0   r  r  r  r  _s         r1   __init__ poisson_binomial_frozen.__init__  s    		 NN@T%=%=%?@	 %4$;$;GW$M		!&7&?&?&Q		# + 3 3GW E		yy,,d;d;1//8r5   Nc                     U R                   R                  " U R                  0 U R                  D6u  pgnU R                   R                  " XR                  XrX440 UD6$ r7   )r  r  r  r  expect)	r0   funclbubconditionalr  rF   r  scales	            r1   r  poisson_binomial_frozen.expect  sK    		--tyyFDIIF yyii"RTRRr5   )rF   r  r   r  r  )NNNF)r   r   r   r   r  r  r   r   r5   r1   r  r    s    9Sr5   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      /$ )Nr  Fr   r   r  r,   r/   s    r1   r2   skellam_gen._shape_info  s:    5%!RVVnE5%!RVVnEG 	Gr5   Nc                 L    UnUR                  X5      UR                  X%5      -
  $ r7   r  )r0   r  r  r:   r;   r'   s         r1   r<   skellam_gen._rvs  s-    $$S,$$S,- 	.r5   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  rT   	_ncx2_pdfr0   rN   r  r  pxs        r1   rW   skellam_gen._pmf  s    [[h'!a%--#q!A#w#>q@--#q!A#w#>q@BB (
 	 ('
 	s   A&B
Bc                 8   [        U5      n[        R                  " SS9   [        R                  " US:  [        R
                  " SU-  SU-  SU-  5      S[        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  rT   	_ncx2_cdfr  s        r1   r\   skellam_gen._cdf  s    !H[[h'!a%--#r!tQsU;cmmAcE1ac7AcEBBDB ( 		 (' 	s   A B


Bc                 F    X-
  nX-   nU[        US-  5      -  nSU-  nX4XV4$ )Nr   r   r0  )r0   r  r  r(  rw   rx   ry   s          r1   r   skellam_gen._stats  s6    yiD#N"W"  r5   r   ro   )r   r   r   r   r   r2   r<   rW   r\   r   r   r   r5   r1   r  r    s!    :G.
!r5   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      /$ )NalphaFr   r   r,   r/   s    r1   r2   yulesimon_gen._shape_info  s    7EArvv;GHHr5   Nc           	          UR                  U5      nUR                  U5      n[        U* [        [        U* U-  5      * 5      -  5      nU$ r7   )standard_exponentialr   r   r   )r0   r
  r:   r;   E1E2anss          r1   r<   yulesimon_gen._rvs  sK    ..t4..t4B3RC%K 00112
r5   c                 :    U[         R                  " XS-   5      -  $ rJ   r   r   r0   rN   r
  s      r1   rW   yulesimon_gen._pmf  s    w||Aqy111r5   c                     US:  $ r   r   )r0   r
  s     r1   rB   yulesimon_gen._argcheck  s    	r5   c                 L    [        U5      [        R                  " XS-   5      -   $ rJ   r   r   r   r  s      r1   rQ   yulesimon_gen._logpmf  s    5zGNN1ai888r5   c                 @    SU[         R                  " XS-   5      -  -
  $ rJ   r  r  s      r1   r\   yulesimon_gen._cdf  s    1w||Aqy1111r5   c                 :    U[         R                  " XS-   5      -  $ rJ   r  r  s      r1   ra   yulesimon_gen._sf  s    7<<19---r5   c                 L    [        U5      [        R                  " XS-   5      -   $ rJ   r  r  s      r1   rX  yulesimon_gen._logsf   s    1vq!)444r5   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$ )	Nr   r   rq   r   r:     1      )r-   rG  r.   nanr   )r0   r
  rv   r  rx   ry   s         r1   r   yulesimon_gen._stats#  sQ   XXeqj"&&%19*=>hhuqyaxECKEAI>#ABvv hhuz2663/XXeai519oQ6%19:MNff XXeqj"&&"-XXeaiaiBMBJ$>$C$)QY$7519$E$G Hff XXeqj"&&"-r5   r   ro   )r   r   r   r   r   r2   r<   rW   rB   rQ   r\   ra   rX  r   r   r   r5   r1   r  r    s6     BI292.5r5   r  	yulesimon)r   rF   c                   H    \ 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)_nchypergeom_geni8  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      /$ )
Nrj  Tr   r(   r'   rk  oddsFr   r,   r/   s    r1   r2   _nchypergeom_gen._shape_infoB  sf    3q"&&k=A3q"&&k=A3q"&&k=A651bff+~FH 	Hr5   c                 v    XUp%nX5-
  n[         R                  " SX&-
  5      n[         R                  " X%5      nXx4$ r   rs  )	r0   rj  r'   rk  r+  rK  rJ  x_minx_maxs	            r1   rG   _nchypergeom_gen._get_supportH  s:    qV

1af%

1!|r5   c                    [         R                  " U5      [         R                  " U5      p![         R                  " U5      [         R                  " U5      pCUR                  [        5      U:H  US:  -  nUR                  [        5      U:H  US:  -  nUR                  [        5      U:H  US:  -  nUS:  nX1:*  n	X!:*  n
XV-  U-  U-  U	-  U
-  $ r   )r-   r  r   r  )r0   rj  r'   rk  r+  cond1cond2cond3cond4cond5cond6s              r1   rB   _nchypergeom_gen._argcheckO  s    zz!}bjjm1**Q-D!14#!#Q/#!#Q/#!#Q/q}u$u,u4u<<r5   c           	      2   ^  [         U 4S j5       nU" XX4XVS9$ )Nc                    > [         R                  " U5      n[        5       n[        UT
R                  5      nU" X!XXe5      n	U	R                  U5      n	U	$ r7   )r-   prodr    getattrrvs_namereshape)rj  r'   rk  r+  r:   r;   lengthurnrv_genr  r0   s             r1   r  $_nchypergeom_gen._rvs.<locals>._rvs1\  sI    WWT]F#%CS$--0Fq=C++d#CJr5   r   r  )r0   rj  r'   rk  r+  r:   r;   r  s   `       r1   r<   _nchypergeom_gen._rvsZ  s&    	#	 
$	 Q1IIr5   c                    ^  [         R                  " XX4U5      u  pp4nUR                  S:X  a  [         R                  " U5      $ [         R                  U 4S j5       nU" XX4U5      $ )Nr   c                 L   > TR                  X2XS5      nUR                  U 5      $ Ng-q=)r  probability)rN   rj  r'   rk  r+  r@  r0   s         r1   _pmf1$_nchypergeom_gen._pmf.<locals>._pmf1m  s$    ))A!51C??1%%r5   )r-   r   r:   
empty_likerQ  )r0   rN   rj  r'   rk  r+  rH  s   `      r1   rW   _nchypergeom_gen._pmfg  s_    ..qQ4@aD66Q;==##		& 
	& Q1&&r5   c                 x   ^  [         R                  U 4S j5       nSU;   d  SU;   a	  U" XX45      OSu  pxSu  pXxX4$ )Nc                 J   > TR                  X!XS5      nUR                  5       $ rF  )r  ru   )rj  r'   rk  r+  r@  r0   s        r1   	_moments1*_nchypergeom_gen._stats.<locals>._moments1v  s!    ))A!51C;;= r5   r  vro   )r-   rQ  )r0   rj  r'   rk  r+  ru   rN  r  rP  rp   rO   s   `          r1   r   _nchypergeom_gen._statst  sL    		! 
	! .1G^sg~	!(! 	Qzr5   r   ro   )r   r   r   r   r   r=  r  r2   rG   rB   r<   rW   r   r   r   r5   r1   r)  r)  8  s3     HDH	=J'
r5   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   r=  r   r  r   r   r5   r1   rS  rS    s    GR H%Dr5   rS  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   r=  r   r  r   r   r5   r1   rW  rW    s    GR H'Dr5   rW  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)r   N)r   r   )r   )j	functoolsr   scipyr   scipy.specialr   r   r   r   rK   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_ufuncsrT   r#   r   r   r   r   r   r   r  r  r=  r?  re  rh  r  r  r  r  r  r  r  r  r  r  r4  r6  rR  rW  r\  rl  ru  ry  r{  r  r  r.   r  r  r  r  r  r  r  r  r  r  r  r  r  r'  r)  rS  rU  rW  rY  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r   r5   r1   <module>rm     s  
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