+
    nDj                       ^ RI Ht ^ RIHt ^ RIHt ^ RIHtHtH	t	H
t ^ RIHu Ht ^ RIHtHt ^ RIHt ^ RIHu Ht ^ RIHt ^ R	IHtHtHtHtHtH t H!t!H"t"H#t#H$t$ ^ RIt%^R
I&H't'H(t(H)t)H*t*H+t+H,t, ^RI-H.t.H/t/H0t0  ! R R]'4      t1]1! RR7      t2 ! R R]14      t3]3! ^RR7      t4 ! R R]'4      t5]5! RR7      t6 ! R R]'4      t7]7! RR7      t8 ! R R]'4      t9]9! RR7      t: ! R R]'4      t;];! ^RR R!7      t< ! R" R#]'4      t=]=! R$R7      t> ! R% R&]'4      t?]?! R'R7      t@ ! R( R)]'4      tA]A! ^R*R+R!7      tB ! R, R-]'4      tC]C! R.R/R07      tD ! R1 R2]'4      tE]E! ^ R3R4R!7      tF ! R5 R6]'4      tG]G! R7^ R8R97      tH ! R: R;]'4      tI]I! R<R=R07      tJ ! R> R?]'4      tK]K! ^R@RAR!7      tL ! RB RC]'4      tM]M! ^RDRER!7      tN ! RF RG]'4      tO]O! ]%P                  ) RHRIR!7      tQ ! RJ RK]'4      tR]R! RLRMRNRO7      tSRgRP ltTRhRQ ltURiRR ltV]S]RutWtX]TP                  ]W]X4      ]SnT        ]UP                  ]W]X4      ]SnU        ]VP                  ]W]X4      ]SnV         ! RS RT],4      tZ ! RU RV]'4      t[][! ]%P                  ) RWRXR!7      t\ ! RY RZ]'4      t]]]! R[^R\7      t^ ! R] R^]'4      t_ ! R_ R`]_4      t`]`! RaRbR07      ta ! Rc Rd]_4      tb]b! ReRfR07      tc]d! ]e! 4       P                  4       P                  4       4      th](! ]h]'4      w  titj]i]j,           tkR# )j    )partial
MethodType)special)entr	logsumexpbetalngammalnN)_poisson_binom_pmf_poisson_binom_cdf)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_PyStochasticLib3c                   n   a  ] tR t^t o RtR tRR ltR tR tR t	R t
R	 tR
 tR tR tRR ltRtV tR# )	binom_gena  
A binomial discrete random variable.

%(before_notes)s

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

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
c                Z    \        R R^ \        P                  3R4      \        RRRR4      .# nTFpTFr      TTr   npinfselfs   &V/data/cameron/venvs/s3viz/lib/python3.14/site-packages/scipy/stats/_discrete_distns.py_shape_infobinom_gen._shape_infoA   0    3q"&&k=A3v|<> 	>    Nc                    \         P                  ! V\         P                  ! V4      8H  4      '       g   \        R 4      hVP	                  \         P
                  ! V\        R7      W#4      # z`n` must be integral.dtype)r-   allr   
ValueErrorbinomialasarrayintr0   r&   r'   sizerandom_states   &&&&&r1   _rvsbinom_gen._rvsE   sH    vva288A;&''455$$RZZ%=qGGr5   c                T    V^ 8  \        V4      ,          V^ 8  ,          V^8*  ,          # r   r   r0   r&   r'   s   &&&r1   	_argcheckbinom_gen._argcheckJ   s'    Q+a.(AF3qAv>>r5   c                    V P                   V3# NarG   s   &&&r1   _get_supportbinom_gen._get_supportM   s    vvqyr5   c                $   \        V4      p\        V^,           4      \        V^,           4      \        W$,
          ^,           4      ,           ,
          pV\        P                  ! WC4      ,           \        P                  ! W$,
          V) 4      ,           # r*   )r   gamlnr   xlogyxlog1py)r0   xr&   r'   kcombilns   &&&&  r1   _logpmfbinom_gen._logpmfP   s\    !H1:qseACEl!:;q,,wqsQB/GGGr5   c                0    \         P                  ! WV4      # rK   )scu
_binom_pmfr0   rU   r&   r'   s   &&&&r1   _pmfbinom_gen._pmfU   s    ~~aA&&r5   c                F    \        V4      p\        P                  ! WBV4      # rK   )r   r[   
_binom_cdfr0   rU   r&   r'   rV   s   &&&& r1   _cdfbinom_gen._cdfY       !H~~aA&&r5   c                F    \        V4      p\        P                  ! WBV4      # rK   )r   r[   	_binom_sfrb   s   &&&& r1   _sfbinom_gen._sf]   s    !H}}Q1%%r5   c                0    \         P                  ! WV4      # rK   )r[   
_binom_isfr]   s   &&&&r1   _isfbinom_gen._isfa       ~~aA&&r5   c                0    \         P                  ! WV4      # rK   )r[   
_binom_ppfr0   qr&   r'   s   &&&&r1   _ppfbinom_gen._ppfd   rn   r5   c                   W,          pWA\         P                  ! V4      ,          ,
          pR R rvRV9   dh   V\         P                  ! V4      ,
          p\         P                  ! W,          4      p	\         P                  ! V	4      p
RV,          V	,          pW,
          pRV9   dM   V\         P                  ! V4      ,
          pW,          p\         P                  ! V4      p
RV,          pW,
          pWEWg3# )Ns       @rV         @)r-   squarer   
reciprocal)r0   r&   r'   momentsmuvarg1g2pqnpq_sqrtt1t2npqs   &&&&         r1   _statsbinom_gen._statsg   s    Uryy|##tB'>RYYq\!BwwqvHx(B'X%BB'>RYYq\!B&Cs#BQBBr5    NNmv)__name__
__module____qualname____firstlineno____doc__r2   rB   rH   rN   rX   r^   rc   rh   rl   rs   r   __static_attributes____classdictcell____classdict__s   @r1   r#   r#      sI     "F>H
?H
''&'' r5   r#   binom)namec                   p   a  ] tR t^}t o RtR tRR ltR tR tR t	R t
R	 tR
 tR tR tR tR tRtV tR# )bernoulli_gena  A Bernoulli discrete random variable.

%(before_notes)s

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

.. math::

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

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

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

%(after_notes)s

%(example)s

c                     \        R RRR4      .# r'   Fr)   r+   r   r/   s   &r1   r2   bernoulli_gen._shape_info       3v|<==r5   Nc                4    \         P                  V ^WVR7      # )r*   r@   rA   )r#   rB   r0   r'   r@   rA   s   &&&&r1   rB   bernoulli_gen._rvs   s    ~~dAq,~OOr5   c                     V^ 8  V^8*  ,          # rE   r   r0   r'   s   &&r1   rH   bernoulli_gen._argcheck   s    Q16""r5   c                2    V P                   V P                  3# rK   )rM   br   s   &&r1   rN   bernoulli_gen._get_support   s    vvtvv~r5   c                0    \         P                  V^V4      # rQ   )r   rX   r0   rU   r'   s   &&&r1   rX   bernoulli_gen._logpmf   s    }}Q1%%r5   c                0    \         P                  V^V4      # rQ   )r   r^   r   s   &&&r1   r^   bernoulli_gen._pmf   s     zz!Q""r5   c                0    \         P                  V^V4      # rQ   )r   rc   r   s   &&&r1   rc   bernoulli_gen._cdf       zz!Q""r5   c                0    \         P                  V^V4      # rQ   )r   rh   r   s   &&&r1   rh   bernoulli_gen._sf   s    yyAq!!r5   c                0    \         P                  V^V4      # rQ   )r   rl   r   s   &&&r1   rl   bernoulli_gen._isf   r   r5   c                0    \         P                  V^V4      # rQ   )r   rs   )r0   rr   r'   s   &&&r1   rs   bernoulli_gen._ppf   r   r5   c                .    \         P                  ^V4      # rQ   )r   r   r   s   &&r1   r   bernoulli_gen._stats   s    ||Aq!!r5   c                F    \        V4      \        ^V,
          4      ,           # rQ   )r   r   s   &&r1   _entropybernoulli_gen._entropy   s    Awac""r5   r   r   )r   r   r   r   r   r2   rB   rH   rN   rX   r^   rc   rh   rl   rs   r   r   r   r   r   s   @r1   r   r   }   sL     0>P#&#
#"##"# #r5   r   	bernoulli)r   r   c                   V   a  ] tR t^t o RtR tRR ltR tR tR t	R t
RR	 ltR
tV tR# )betabinom_gena  
A beta-binomial discrete random variable.

%(before_notes)s

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

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

%(example)s
c                    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# r&   TFrM   r   r(   FFr,   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                    VP                  W#V4      p\        P                  ! V\        P                  ! V4      8H  4      '       g   \	        R 4      hVP                  \        P                  ! V\        R7      Wd4      # r7   )betar-   r:   r   r;   r<   r=   r>   r0   r&   rM   r   r@   rA   r'   s   &&&&&& r1   rB   betabinom_gen._rvs   sZ    aD)vva288A;&''455$$RZZ%=qGGr5   c                
    ^ V3# rE   r   r0   r&   rM   r   s   &&&&r1   rN   betabinom_gen._get_support       !tr5   c                T    V^ 8  \        V4      ,          V^ 8  ,          V^ 8  ,          # rE   rF   r   s   &&&&r1   rH   betabinom_gen._argcheck   '    Q+a.(AE2a!e<<r5   c                    \        V4      p\        V^,           4      ) \        W%,
          ^,           V^,           4      ,
          pV\        WS,           W%,
          V,           4      ,           \        W44      ,
          # rQ   )r   r   r	   r0   rU   r&   rM   r   rV   rW   s   &&&&&  r1   rX   betabinom_gen._logpmf   sS    !Hq1u:+quqy!a% 88quqy11F1L@@r5   c                8    \        V P                  WW44      4      # rK   r   rX   r0   rU   r&   rM   r   s   &&&&&r1   r^   betabinom_gen._pmf       4<<a+,,r5   c                L   W"V,           ,          p^V,
          pW,          pWV,           V,           ,          V,          V,          W#,           ^,           ,          pRRrRV9   d`   R\        V4      ,          p	WV,           ^V,          ,           W2,
          ,          ,          p	WV,           ^,           W#,           ,          ,          p	RV9   EdY   W#,           P                  VP                  4      p
WV,           ^,
          ^V,          ,           ,          p
V
^V,          V,          V^,
          ,          ,          p
V
^V^,          ,          ,          p
V
^V,          V,          V,          ^V,
          ,          ,          p
V
^V,          V,          V^,          ,          ,          p
WV,           ^,          ^V,           V,           ,          ,          p
WV,          V,          W#,           ^,           ,          W#,           ^,           ,          W#,           V,           ,          ,          p
V
^,          p
WxW3# )r*   Nrv         ?rV   )r   astyper9   )r0   r&   rM   r   r{   e_pe_qr|   r}   r~   r   s   &&&&&      r1   r   betabinom_gen._stats   s   q5k#gWq519o#c)QUQY7tB'>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   r   r   )r   r   r   r   r   r2   rB   rN   rH   rX   r^   r   r   r   r   s   @r1   r   r      s6     "FE
H=A
- r5   r   	betabinomc                   j   a  ] tR tRt o RtR tRR ltR tR tR t	R	 t
R
 tR tR tR tR tRtV tR# )
nbinom_geni  a?  
A negative binomial discrete random variable.

%(before_notes)s

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

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
c                Z    \        R R^ \        P                  3R4      \        RRRR4      .# r%   r,   r/   s   &r1   r2   nbinom_gen._shape_infoS  r4   r5   Nc                &    VP                  WV4      # rK   )negative_binomialr?   s   &&&&&r1   rB   nbinom_gen._rvsW  s    --aD99r5   c                4    V^ 8  V^ 8  ,          V^8*  ,          # rE   r   rG   s   &&&r1   rH   nbinom_gen._argcheckZ  s    A!a% AF++r5   c                0    \         P                  ! WV4      # rK   )r[   _nbinom_pmfr]   s   &&&&r1   r^   nbinom_gen._pmf]  s    qQ''r5   c                    \        W!,           4      \        V^,           4      ,
          \        V4      ,
          pWB\        V4      ,          ,           \        P                  ! W) 4      ,           # rQ   )rR   r   r   rT   )r0   rU   r&   r'   coeffs   &&&& r1   rX   nbinom_gen._logpmfa  sD    ac
U1Q3Z'%(2Qx'//!R"888r5   c                F    \        V4      p\        P                  ! WBV4      # rK   )r   r[   _nbinom_cdfrb   s   &&&& r1   rc   nbinom_gen._cdfe  s    !HqQ''r5   c                |   \        V4      p\        P                  ! WBV4      w  rBpV P                  WBV4      pVR 8  pR pTp\        P                  ! RR7      ;_uu_ 4        V! WF,          W&,          W6,          4      W&   \        P
                  ! WV( ,          4      W( &   RRR4       V#   + '       g   i     T# ; i)      ?c                 x    \         P                  ! \        P                  ! V ^,           V^V,
          4      ) 4      # rQ   )r-   r   r   betainc)rV   r&   r'   s   &&&r1   f1nbinom_gen._logcdf.<locals>.f1n  s)    88W__QUAq1u==>>r5   ignore)divideN)r   r-   broadcast_arraysrc   errstater   )	r0   rU   r&   r'   rV   cdfcondr   logcdfs	   &&&&     r1   _logcdfnbinom_gen._logcdfi  s    !H%%aA.aiia Sy	? [[))agqw8FLFF3u:.F5M *  *) s   !?B**B;	c                F    \        V4      p\        P                  ! WBV4      # rK   )r   r[   
_nbinom_sfrb   s   &&&& r1   rh   nbinom_gen._sfx  re   r5   c                    \         P                  ! R R7      ;_uu_ 4        \        P                  ! WV4      uuRRR4       #   + '       g   i     R# ; ir   overN)r-   r   r[   _nbinom_isfr]   s   &&&&r1   rl   nbinom_gen._isf|  .    [[h''??1+ ('''   AA	c                    \         P                  ! R R7      ;_uu_ 4        \        P                  ! WV4      uuRRR4       #   + '       g   i     R# ; ir  )r-   r   r[   _nbinom_ppfrq   s   &&&&r1   rs   nbinom_gen._ppf  r  r  c                    \         P                  ! W4      \         P                  ! W4      \         P                  ! W4      \         P                  ! W4      3# rK   )r[   _nbinom_mean_nbinom_variance_nbinom_skewness_nbinom_kurtosis_excessrG   s   &&&r1   r   nbinom_gen._stats  sD    Q"  &  &''-	
 	
r5   r   r   )r   r   r   r   r   r2   rB   rH   r^   rX   rc   r   rh   rl   rs   r   r   r   r   s   @r1   r   r     sG     9t>:,(9(',,
 
r5   r   nbinomc                   P   a  ] tR tRt o RtR tRR ltR tR tR t	RR	 lt
R
tV tR# )betanbinom_geni  a  
A beta-negative-binomial discrete random variable.

%(before_notes)s

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

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

%(example)s
c                    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# r   r,   r/   s   &r1   r2   betanbinom_gen._shape_info  r   r5   Nc                J    VP                  W#V4      pVP                  WV4      # rK   )r   r   r   s   &&&&&& r1   rB   betanbinom_gen._rvs  s'    aD)--aD99r5   c                T    V^ 8  \        V4      ,          V^ 8  ,          V^ 8  ,          # rE   rF   r   s   &&&&r1   rH   betanbinom_gen._argcheck  r   r5   c                    \        V4      p\        P                  ! W%,           4      ) \        W%^,           4      ,
          pV\        W2,           WE,           4      ,           \        W44      ,
          # rQ   )r   r-   r   r	   r   s   &&&&&  r1   rX   betanbinom_gen._logpmf  sI    !H66!%=.6!U#33qu--q<<r5   c                8    \        V P                  WW44      4      # rK   r   r   s   &&&&&r1   r^   betanbinom_gen._pmf  r   r5   c                   R  p\         P                  ! V^8  WV3V\        P                  R7      pR p\         P                  ! V^8  WV3V\        P                  R7      pRRrR p
RV9   d.   \         P                  ! V^8  WV3V
\        P                  R7      pR pRV9   d.   \         P                  ! V^8  WV3V\        P                  R7      p	WgW3# )c                 .    W,          VR ,
          ,          # r   r   r&   rM   r   s   &&&r1   mean#betanbinom_gen._stats.<locals>.mean  s    5AF##r5   
fill_valuec                     W,          W,           R ,
          ,          W,           R ,
          ,          VR,
          VR ,
          R,          ,          ,          # )r   rw   r   r!  s   &&&r1   r}   "betanbinom_gen._stats.<locals>.var  s;    EQURZ(AEBJ7B1r6B,.0 1r5   Nc                    ^V ,          V,           R,
          ^V,          V,           R,
          ,          VR,
          ,          \        W,          W,           R,
          ,          W!,           R,
          ,          VR,
          ,          4      ,          # )   r         @rw   r   r!  s   &&&r1   skew#betanbinom_gen._stats.<locals>.skew  sf    UQY^A	B72v!%aequrz&:aebj&I2v' "   !r5   rv   c                 T   VR ,
          pVR,
          R ,          VR ,          V^V,          R,
          ,          ,           RVR,
          ,          V,          ,           ,          RV R ,          ,          VR,           VR ,          ,          VR,           VR,
          ,          V,          ,           R VR,
          ^,          ,          ,           ,          ,           ^VR,
          ,          V ,          VR,           VR ,          ,          VR,           VR,
          ,          V,          ,           R VR,
          R ,          ,          ,           ,          ,           pVR,
          VR,
          ,          V,          V ,          W,           R,
          ,          W,           R,
          ,          pW4,          V,          R,
          # )rw   r   rx   r*        @g      @r   )r&   rM   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   rV   xpxapply_wherer-   r.   )r0   r&   rM   r   r{   r"  r|   r}   r~   r   r,  r3  s   &&&&&       r1   r   betanbinom_gen._stats  s    	$__QUQ1ItG	1 ooa!eaAYGtB	! '>Qq	4BFFKB	4 '>Qq	8OBr5   r   r   r   )r   r   r   r   r   r2   rB   rH   rX   r^   r   r   r   r   s   @r1   r  r    s/     #HE
:==
-! !r5   r  
betanbinomc                   j   a  ] tR tRt o RtR tRR ltR tR tR t	R	 t
R
 tR tR tR tR tRtV tR# )geom_geni  a  A geometric discrete random variable.

%(before_notes)s

See Also
--------
planck

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

%(example)s
c                     \        R RRR4      .# r   r   r/   s   &r1   r2   geom_gen._shape_info  r   r5   Nc                    VP                  WR 7      p\        P                  ! VP                  4      P                  p\        P
                  ! V^ 8  WT4      # r@   )	geometricr-   iinfor9   maxwhere)r0   r'   r@   rA   resmax_ints   &&&&  r1   rB   geom_gen._rvs  sD    $$Q$2 ((399%))xxa..r5   c                     V^8*  V^ 8  ,          # rQ   r   r   s   &&r1   rH   geom_gen._argcheck  s    Q1q5!!r5   c                Z    \         P                  ! ^V,
          V^,
          4      V,          # rQ   )r-   powerr0   rV   r'   s   &&&r1   r^   geom_gen._pmf  s     xx!QqS!A%%r5   c                `    \         P                  ! V^,
          V) 4      \        V4      ,           # rQ   )r   rT   r   rL  s   &&&r1   rX   geom_gen._logpmf!  s"    q1uqb)CF22r5   c                R    \        V4      p\        \        V) 4      V,          4      ) # rK   )r   r   r   r0   rU   r'   rV   s   &&& r1   rc   geom_gen._cdf$  s#    !HeQBik"""r5   c                L    \         P                  ! V P                  W4      4      # rK   )r-   r   _logsfr   s   &&&r1   rh   geom_gen._sf(  s    vvdkk!'((r5   c                >    \        V4      pV\        V) 4      ,          # rK   )r   r   rQ  s   &&& r1   rT  geom_gen._logsf+  s    !Hr{r5   c                    \        \        V) 4      \        V) 4      ,          4      pV P                  V^,
          V4      p\        P                  ! WA8  V^ 8  ,          V^,
          V4      # rQ   )r   r   rc   r-   rD  )r0   rr   r'   valstemps   &&&  r1   rs   geom_gen._ppf/  sS    E1"Iqb	)*yya#xxtax0$q&$??r5   c                    R V,          pR V,
          pW1,          V,          pRV,
          \        V4      ,          p\        P                  ! . ROV4      R V,
          ,          pW$WV3# )r   rw   )r*   i   )r   r-   polyval)r0   r'   r|   qrr}   r~   r   s   &&     r1   r   geom_gen._stats4  sT    UUfqj!etBxZZ
A&A.r5   c                    \         P                  ! V4      ) \         P                  ! V) 4      R V,
          ,          V,          ,
          # r   )r-   r   r   r   s   &&r1   r   geom_gen._entropy<  s/    q	zBHHaRLCE2Q666r5   r   r   )r   r   r   r   r   r2   rB   rH   r^   rX   rc   rh   rT  rs   r   r   r   r   r   s   @r1   r;  r;    sH     @>/"&3#)@
7 7r5   r;  geomzA geometric)rM   r   longnamec                   p   a  ] tR tRt o RtR tRR ltR tR tR t	R	 t
R
 tR tR tR tR tR tRtV tR# )hypergeom_geniC  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

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

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)
c                    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# )MTr&   Nr(   r,   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                6    VP                  W!V,
          W4R 7      # r?  )hypergeometric)r0   rh  r&   ri  r@   rA   s   &&&&&&r1   rB   hypergeom_gen._rvs  s    **1c1*@@r5   c                v    \         P                  ! W1V,
          ,
          ^ 4      \         P                  ! W#4      3# rE   r-   maximumminimum)r0   rh  r&   ri  s   &&&&r1   rN   hypergeom_gen._get_support  s'    zz!qS'1%rzz!'777r5   c                    V^ 8  V^ 8  ,          V^ 8  ,          pWBV8*  W18*  ,          ,          pV\        V4      \        V4      ,          \        V4      ,          ,          pV# rE   rF   )r0   rh  r&   ri  r   s   &&&& r1   rH   hypergeom_gen._argcheck  sU    A!q&!Q!V,aAF##AQ/+a.@@r5   c                   Y#reWV,
          p\        V^,           ^4      \        V^,           ^4      ,           \        WT,
          ^,           V^,           4      ,           \        V^,           Wa,
          ^,           4      ,
          \        WA,
          ^,           Wt,
          V,           ^,           4      ,
          \        V^,           ^4      ,
          pV# rQ   r	   )	r0   rV   rh  r&   ri  totgoodbadresults	   &&&&&    r1   rX   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    \         P                  ! WWB4      # rK   )r[   _hypergeom_pmfr0   rV   rh  r&   ri  s   &&&&&r1   r^   hypergeom_gen._pmf      !!!--r5   c                0    \         P                  ! WWB4      # rK   )r[   _hypergeom_cdfr  s   &&&&&r1   rc   hypergeom_gen._cdf  r  r5   c                   R V,          R V,          R V,          r2pW,
          pW^,           ,          RV,          W,
          ,          ,
          RV,          V,          ,
          pWQ^,
          V,          V,          ,          pVRV,          V,          W,
          ,          V,          RV,          ^,
          ,          ,          pWRV,          W,
          ,          V,          VR,
          ,          VR,
          ,          ,          p\         P                  ! W#V4      \         P                  ! W#V4      \         P                  ! W#V4      V3# )r   rx   r/  rw   r*  )r[   _hypergeom_mean_hypergeom_variance_hypergeom_skewness)r0   rh  r&   ri  mr   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                    \         P                  W1V,
          ,
          \        W#4      ^,            pV P                  WAW#4      p\         P                  ! \        V4      ^ R7      # )r*   axis)r-   r_minpmfsumr   )r0   rh  r&   ri  rV   rY  s   &&&&  r1   r   hypergeom_gen._entropy  sE    EE!1u+c!i!m,xxa#vvd4jq))r5   c                0    \         P                  ! WWB4      # rK   )r[   _hypergeom_sfr  s   &&&&&r1   rh   hypergeom_gen._sf  s      q,,r5   c                   . p\        \        P                  ! WW44      !   F  w  rgrVR ,           VR ,           ,          VR ,
          V	R ,
          ,          8  d7   VP                  \	        \        V P                  WgW4      4      ) 4      4       Km  \        P                  ! V^,           V	^,           4      p
VP                  \        V P                  WW4      4      4       K  	  \        P                  ! V4      # r   )zipr-   r   appendr   r   r   aranger   rX   r=   r0   rV   rh  r&   ri  rE  quantrx  ry  drawk2s   &&&&&      r1   rT  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                   . p\        \        P                  ! WW44      !   F  w  rgrVR ,           VR ,           ,          VR ,
          V	R ,
          ,          8  d7   VP                  \	        \        V P                  WgW4      4      ) 4      4       Km  \        P                  ! ^ V^,           4      p
VP                  \        V P                  WW4      4      4       K  	  \        P                  ! V4      # r  )r  r-   r   r  r   r   logsfr  r   rX   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   r   )r   r   r   r   r   r2   rB   rN   rH   rX   r^   rc   r   r   rh   rT  r   r   r   r   s   @r1   rf  rf  C  sO     GPC
A8..
"*
-

 
r5   rf  	hypergeomc                   R   a  ] tR tRt o RtR tR tR tRR ltR t	R	 t
R
 tRtV tR# )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

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

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

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

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
c                    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# )rh  Tr&   rr(   r,   r/   s   &r1   r2   nhypergeom_gen._shape_infoD  rk  r5   c                
    ^ V3# rE   r   )r0   rh  r&   r  s   &&&&r1   rN   nhypergeom_gen._get_supportI  r   r5   c                    V^ 8  W!8*  ,          V^ 8  ,          W1V,
          8*  ,          pV\        V4      \        V4      ,          \        V4      ,          ,          pV# rE   rF   )r0   rh  r&   r  r   s   &&&& r1   rH   nhypergeom_gen._argcheckL  sK    Q16"a1f-c:AQ/+a.@@r5   Nc                8   a  \         V 3R  l4       pV! WW4VR7      # )c                 (  < SP                  WV4      w  rV\        P                  ! WV^,           4      pSP                  WpW4      p\	        WRRR7      p	V	! VP                  VR7      4      P                  \        4      p
Vf   V
P                  4       # V
# )r*   nextextrapolate)kindr%  r@  )	supportr-   r  r   r   uniformr   r>   item)rh  r&   r  r@   rA   rM   r   ksr   ppfrvsr0   s   &&&&&      r1   _rvs1"nhypergeom_gen._rvs.<locals>._rvs1S  s     <<a(DA1c"B((2!'C3MJCl***56==cBC|xxz!Jr5   r   r   )r0   rh  r&   r  r@   rA   r  s   f&&&&& r1   rB   nhypergeom_gen._rvsQ  s&    	#		 
$		 Q1lCCr5   c                V    \         P                  ! V^ 8g  V^ 8g  ,          WW43R RR7      # )r   c                    \        V ^,           V4      ) \        W,           ^4      ,           \        W ,
          ^,           W,
          V,
          ^,           4      ,
          \        W,
          V ,
          ^,           ^4      ,           \        V^,           W,
          ^,           4      ,           \        V^,           ^4      ,
          # rQ   rw  )rV   rh  r&   r  s   &&&&r1   <lambda>(nhypergeom_gen._logpmf.<locals>.<lambda>d  s    1a.6!#q>1!#a%Qq)*,213q57A,>?!A#qs1u%&(.qsA7r5           r$  )r6  r7  r0   rV   rh  r&   r  s   &&&&&r1   rX   nhypergeom_gen._logpmfa  s3    !VQ!8  	r5   c                8    \        V P                  WW44      4      # rK   r   r  s   &&&&&r1   r^   nhypergeom_gen._pmfj  s     4<<a+,,r5   c                <   R V,          R V,          R V,          r2pW2,          W,
          ^,           ,          pW1^,           ,          V,          W,
          ^,           W,
          ^,           ,          ,          ^W1V,
          ^,           ,          ,
          ,          pRRrvWEWg3# )r   Nr   )r0   rh  r&   r  r|   r}   r~   r   s   &&&&    r1   r   nhypergeom_gen._statso  sv     Q$1bdaSACE]1gaiACEACE?+q1!A;? tBr5   r   r   )r   r   r   r   r   r2   rN   rH   rB   rX   r^   r   r   r   r   s   @r1   r  r    s6     aFC

D -
 r5   r  
nhypergeomc                   L   a  ] tR tRt o RtR tRR ltR tR tR t	R	 t
R
tV tR# )
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                     \        R RRR4      .# r   r   r/   s   &r1   r2   logser_gen._shape_info  r   r5   Nc                &    VP                  WR 7      # r?  )	logseriesr   s   &&&&r1   rB   logser_gen._rvs  s     %%a%33r5   c                     V^ 8  V^8  ,          # rE   r   r   s   &&r1   rH   logser_gen._argcheck  s    A!a%  r5   c                    \         P                  ! W!4      ) R ,          V,          \        P                  ! V) 4      ,          # r   )r-   rK  r   r   rL  s   &&&r1   r^   logser_gen._pmf  s,    $q(7==!+<<<r5   c                    R p\         P                  ! V^,           W24      ) \         P                  ! V^,           V4      ,          \        P                  ! V) 4      ,          # )g0.++)r   r   r   r-   r   )r0   rV   r'   tinys   &&& r1   rh   logser_gen._sf  sF    
 !T--QqS$0GG"((TUSU,VVr5   c                P   \         P                  ! V) 4      pWR ,
          ,          V,          pV) V,          VR ,
          ^,          ,          pWCV,          ,
          pV) V,          R V,           ,          R V,
          ^,          ,          pV^V,          V,          ,
          ^V^,          ,          ,           pV\        P                  ! VR4      ,          pV) V,          R V^,
          ^,          ,          ^V,          V^,
          ^,          ,          ,
          ^V,          V,          V^,
          ^,          ,          ,           ,          p	V	^V,          V,          ,
          ^V,          V,          V,          ,           ^V^,          ,          ,
          p
W^,          ,          R,
          pW5W3# )r         ?r*  )r   r   r-   rK  )r0   r'   r  r|   mu2pr}   mu3pmu3r~   mu4pmu4r   s   &&          r1   r   logser_gen._stats  s4   MM1"c']QrAvS1$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   r   )r   r   r   r   r   r2   rB   rH   r^   rh   r   r   r   r   s   @r1   r  r    s.     0>4
!=W r5   r  logserzA logarithmicc                   ^   a  ] tR tRt o RtR tR tRR ltR tR t	R	 t
R
 tR tR tRtV tR# )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                @    \        R R^ \        P                  3R4      .# )r|   Fr(   r,   r/   s   &r1   r2   poisson_gen._shape_info  s    4BFF]CDDr5   c                    V^ 8  # rE   r   )r0   r|   s   &&r1   rH   poisson_gen._argcheck  s    Qwr5   Nc                $    VP                  W4      # rK   poisson)r0   r|   r@   rA   s   &&&&r1   rB   poisson_gen._rvs  s    ##B--r5   c                n    \         P                  ! W4      \        V^,           4      ,
          V,
          pV# rQ   )r   rS   rR   )r0   rV   r|   Pks   &&& r1   rX   poisson_gen._logpmf  s'    ]]1!E!a%L025	r5   c                6    \        V P                  W4      4      # rK   r   )r0   rV   r|   s   &&&r1   r^   poisson_gen._pmf  s    4<<&''r5   c                D    \        V4      p\        P                  ! W24      # rK   )r   r   pdtrr0   rU   r|   rV   s   &&& r1   rc   poisson_gen._cdf  s    !H||A""r5   c                D    \        V4      p\        P                  ! W24      # rK   )r   r   pdtrcr  s   &&& r1   rh   poisson_gen._sf  s    !H}}Q##r5   c                    \        \        P                  ! W4      4      p\        P                  ! V^,
          ^ 4      p\        P
                  ! WB4      p\        P                  ! WQ8  WC4      # rQ   )r   r   pdtrikr-   rq  r  rD  )r0   rr   r|   rY  vals1rZ  s   &&&   r1   rs   poisson_gen._ppf  sJ    GNN1)*

4!8Q'||E&xx	5//r5   c                    Tp\         P                  ! V4      pV^ 8  p\        P                  ! WCR \         P                  R7      p\        P                  ! WCR \         P                  R7      pWWV3# )r   c                 &    \        R V ,          4      # r   r+  rU   s   &r1   r  $poisson_gen._stats.<locals>.<lambda>  s    SUr5   r$  c                     R V ,          # r   r   r  s   &r1   r  r    s    Ar5   )r-   r=   r6  r7  r.   )r0   r|   r}   tmp
mu_nonzeror~   r   s   &&     r1   r   poisson_gen._stats  sW    jjn1W
__Z.CPRPVPVW__Zo"&&Qr5   r   r   )r   r   r   r   r   r2   rH   rB   rX   r^   rc   rh   rs   r   r   r   r   s   @r1   r  r    s=     0E.(#$0 r5   r  r  z	A Poisson)r   rd  c                   d   a  ] tR tRt o RtR tR tR tR tR t	R t
R	 tRR ltR tR tRtV tR
# )
planck_geni	  a  A Planck discrete exponential random variable.

%(before_notes)s

See Also
--------
geom

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

%(example)s
c                @    \        R R^ \        P                  3R4      .# )lambda_Fr   r,   r/   s   &r1   r2   planck_gen._shape_info$  s    9ea[.IJJr5   c                    V^ 8  # rE   r   )r0   r  s   &&r1   rH   planck_gen._argcheck'  s    {r5   c                L    \        V) 4      ) \        V) V,          4      ,          # rK   )r   r   )r0   rV   r  s   &&&r1   r^   planck_gen._pmf*  s     whWHQJ//r5   c                N    \        V4      p\        V) V^,           ,          4      ) # rQ   )r   r   r0   rU   r  rV   s   &&& r1   rc   planck_gen._cdf-  s#    !Hwh!n%%%r5   c                6    \        V P                  W4      4      # rK   )r   rT  )r0   rU   r  s   &&&r1   rh   planck_gen._sf1  s    4;;q*++r5   c                :    \        V4      pV) V^,           ,          # rQ   r   r  s   &&& r1   rT  planck_gen._logsf4  s    !Hx1~r5   c                    \        RV,          \        V) 4      ,          ^,
          4      pV^,
          P                  ! V P                  V4      !  pV P	                  WB4      p\
        P                  ! WQ8  WC4      # )r         )r   r   cliprN   rc   r-   rD  )r0   rr   r  rY  r  rZ  s   &&&   r1   rs   planck_gen._ppf8  s^    DL5!9,Q./a 1 1' :<yy(xx	5//r5   Nc                N    \        V) 4      ) pVP                  WBR 7      R,
          # )r@  r   )r   rA  )r0   r  r@   rA   r'   s   &&&& r1   rB   planck_gen._rvs>  s)    G8_%%a%3c99r5   c                    ^\        V4      ,          p\        V) 4      \        V) 4      ^,          ,          p^\        VR,          4      ,          p^^\        V4      ,          ,           pW#WE3# r*   rw   )r   r   r   )r0   r  r|   r}   r~   r   s   &&    r1   r   planck_gen._statsC  sZ    uW~7(mUG8_q00tGCK  qgr5   c                p    \        V) 4      ) pV\        V) 4      ,          V,          \        V4      ,
          # rK   )r   r   r   )r0   r  Cs   && r1   r   planck_gen._entropyJ  s/    G8_sG8}$Q&Q//r5   r   r   )r   r   r   r   r   r2   rH   r^   rc   rh   rT  rs   rB   r   r   r   r   r   s   @r1   r  r  	  sB     4K0&,0:
0 0r5   r  planckzA discrete exponential c                   N   a  ] tR tRt o RtR tR tR tR tR t	R t
R	 tR
tV tR# )boltzmann_geniR  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    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )r  Fri  Tr   r,   r/   s   &r1   r2   boltzmann_gen._shape_infoh  s:    9ea[.I3q"&&k>BD 	Dr5   c                @    V^ 8  V^ 8  ,          \        V4      ,          # rE   rF   r0   r  ri  s   &&&r1   rH   boltzmann_gen._argcheckl  s    !A&Q77r5   c                ,    V P                   V^,
          3# rQ   rL   r$  s   &&&r1   rN   boltzmann_gen._get_supporto  s    vvq1u}r5   c                    ^\        V) 4      ,
          ^\        V) V,          4      ,
          ,          pV\        V) V,          4      ,          # rQ   r   )r0   rV   r  ri  facts   &&&& r1   r^   boltzmann_gen._pmfr  s<     #wh-!C
O"34C
O##r5   c                    \        V4      p^\        V) V^,           ,          4      ,
          ^\        V) V,          4      ,
          ,          # rQ   )r   r   )r0   rU   r  ri  rV   s   &&&& r1   rc   boltzmann_gen._cdfx  s9    !H#wh!n%%#whqj/(9::r5   c                H   V^\        V) V,          4      ,
          ,          p\        RV,          \        ^V,
          4      ,          ^,
          4      pV^,
          P                  R\        P
                  4      pV P                  WbV4      p\        P                  ! Wq8  We4      # )r*   r  r  )r   r   r   r  r-   r.   rc   rD  )r0   rr   r  ri  qnewrY  r  rZ  s   &&&&    r1   rs   boltzmann_gen._ppf|  sv    !C
O#$DL3qv;.q01ac266*yy+xx	5//r5   c                "   \        V) 4      p\        V) V,          4      pVR V,
          ,          W$,          ^V,
          ,          ,
          pVR V,
          ^,          ,          W",          V,          ^V,
          ^,          ,          ,
          p^V,
          ^V,
          ,          pW7^,          ,          W",          V,          ,
          pV^V,           ,          V^,          ,          V^,          V,          ^V,           ,          ,
          p	WR,          ,          p	V^^V,          ,           W3,          ,           ,          V^,          ,          V^,          V,          ^^V,          ,           WD,          ,           ,          ,
          p
W,          V,          p
WVW3# )r   r  r)  )r0   r  ri  zzNr|   r}   trmtrm2r~   r   s   &&&        r1   r   boltzmann_gen._stats  s   M'!_AYqtQrT{"Q
lQSVQrTAI--taclq&13r6!!WS!V^ad2gqtn,+!A#ac	]36!AqD2Iq2vbe|$<<Yr5   r   N)r   r   r   r   r   r2   rH   rN   r^   rc   rs   r   r   r   r   s   @r1   r   r   R  s3     *D8$;0 r5   r   	boltzmannz!A truncated discrete exponential )r   rM   rd  c                   ^   a  ] tR tRt o RtR tR tR tR tR t	R t
R	 tRR ltR tRtV tR
# )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                    \        R R\        P                  ) \        P                  3R4      \        RR\        P                  ) \        P                  3R4      .# )lowThighr   r,   r/   s   &r1   r2   randint_gen._shape_info  sH    5$"&&"&&(9>J64266'266):NKM 	Mr5   c                J    W!8  \        V4      ,          \        V4      ,          # rK   rF   r0   r;  r<  s   &&&r1   rH   randint_gen._argcheck  s    
k#..T1BBBr5   c                    W^,
          3# rQ   r   r?  s   &&&r1   rN   randint_gen._get_support  s    F{r5   c                    \         P                  ! V4      \         P                  ! V\         P                  R 7      V,
          ,          p\         P                  ! W8  W8  ,          VR4      # )r8   r  )r-   	ones_liker=   int64rD  )r0   rV   r;  r<  r'   s   &&&& r1   r^   randint_gen._pmf  sD    LLOrzz$bhh?#EFxxah/B77r5   c                P    \        V4      pWB,
          R ,           W2,
          ,          # r   r  )r0   rU   r;  r<  rV   s   &&&& r1   rc   randint_gen._cdf  s    !H",,r5   c                    \        WV,
          ,          V,           4      ^,
          pV^,
          P                  W#4      pV P                  WRV4      p\        P                  ! Wa8  WT4      # rQ   )r   r  rc   r-   rD  )r0   rr   r;  r<  rY  r  rZ  s   &&&&   r1   rs   randint_gen._ppf  sR    A$s*+a/*yyT*xx	5//r5   c                    \         P                  ! V4      \         P                  ! V4      rCW4,           R ,
          ^,          pW4,
          pWf,          ^,
          R,          pRpRWf,          R ,           ,          Wf,          R ,
          ,          p	WWW3# )r   g      (@r  g333333)r-   r=   )
r0   r;  r<  m2m1r|   dr}   r~   r   s
   &&&       r1   r   randint_gen._stats  sk    D!2::c?Bgmq GsQw$s#qsSy1r5   Nc                   \         P                  ! V4      P                  ^8X  d3   \         P                  ! V4      P                  ^8X  d   \        WAW#R7      # Ve-   \         P                  ! W4      p\         P                  ! W#4      p\         P
                  ! \        \        V4      \         P                  ! \        4      .R7      pV! W4      # )z=An array of *size* random integers >= ``low`` and < ``high``.r@  )otypes)	r-   r=   r@   r   broadcast_to	vectorizer   r9   r>   )r0   r;  r<  r@   rA   randints   &&&&& r1   rB   randint_gen._rvs  s    ::c?1$D)9)>)>!)C4CC
 //#,C??4.D,,w|\B')xx}o7s!!r5   c                $    \        W!,
          4      # rK   )r   r?  s   &&&r1   r   randint_gen._entropy  s    4:r5   r   r   )r   r   r   r   r   r2   rH   rN   r^   rc   rs   r   rB   r   r   r   r   s   @r1   r9  r9    s?     =~MC8
-0"" r5   r9  rT  z#A discrete uniform (random integer)c                   F   a  ] tR tRt o RtR tR
R ltR tR tR t	R	t
V tR# )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                @    \        R R^\        P                  3R4      .# rM   Fr   r,   r/   s   &r1   r2   zipf_gen._shape_info=      3266{NCDDr5   Nc                &    VP                  WR 7      # r?  )zipf)r0   rM   r@   rA   s   &&&&r1   rB   zipf_gen._rvs@  s       ..r5   c                    V^8  # rQ   r   r0   rM   s   &&r1   rH   zipf_gen._argcheckC  s    1ur5   c                    VP                  \        P                  4      pR \        P                  ! V^4      ,          W) ,          ,          pV# r   )r   r-   float64r   zeta)r0   rV   rM   r  s   &&& r1   r^   zipf_gen._pmfF  s7    HHRZZ 7<<1%%2-	r5   c                h    \         P                  ! W!^,           8  W!3R \        P                  R7      # )r*   c                 t    \         P                  ! W,
          ^4      \         P                  ! V ^4      ,          # rQ   )r   rf  )rM   r&   s   &&r1   r   zipf_gen._munp.<locals>.<lambda>O  s!    aeQ/',,q!2DDr5   r$  r5  )r0   r&   rM   s   &&&r1   _munpzipf_gen._munpL  s*    AIvDvv 	r5   r   r   )r   r   r   r   r   r2   rB   rH   r^   rk  r   r   r   s   @r1   rY  rY    s*     )VE/ r5   rY  r_  zA Zipfc                   N   a  ] tR tRt o RtR tR tR tR tR t	R t
R	 tR
tV tR# )zipfian_geniV  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    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )rM   FTr&   r(   r   r,   r/   s   &r1   r2   zipfian_gen._shape_info  s:    3266{MB3q"&&k>BD 	Dr5   c           	         V^ 8  V\         P                  ! \         P                  ! V^R4      4      P                  \         P                  R7      8H  ,          # )r   r8   i)r-   r=   r  r   rE  r0   rM   r&   s   &&&r1   rH   zipfian_gen._argcheck  sG     abjjAy!9:AAAQQS 	Tr5   c                2    ^\         P                  ! V4      3# rQ   )r-   r   rr  s   &&&r1   rN   zipfian_gen._get_support  s    "((1+~r5   c                    \         P                  ! V4      p\         P                  ! V4      p\        P                  ! WW24      # rK   r-   r   r[   _normalized_gen_harmonicr0   rV   rM   r&   s   &&&&r1   r^   zipfian_gen._pmf  s/    HHQKHHQK++A!77r5   c                    \         P                  ! V4      p\         P                  ! V4      p\        P                  ! ^WV4      # rQ   rw  ry  s   &&&&r1   rc   zipfian_gen._cdf  s1    HHQKHHQK++AqQ77r5   c                    \         P                  ! V4      p\         P                  ! V4      p\        P                  ! V^,           W3V4      # rQ   rw  ry  s   &&&&r1   rh   zipfian_gen._sf  s5    HHQKHHQK++AE1;;r5   c                d   \         P                  ! V4      p\        P                  ! W!4      p\        P                  ! W!^,
          4      p\        P                  ! W!^,
          4      p\        P                  ! W!^,
          4      p\        P                  ! W!^,
          4      pWC,          pWS,          V^,          ,
          p	V^,          p
W,          pWc,          ^V,          V,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           VR,          ,          pV^,          V,          ^V^,          ,          V,          V,          ,
          ^V,          V^,          ,          V,          ,           ^V^,          ,          ,
          V	^,          ,          pV^,          pWW3# )r*   r  )r-   r   r[   _gen_harmonic)r0   rM   r&   HnaHna1Hna2Hna3Hna4mu1mu2nmu2dmu2r~   r   s   &&&           r1   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r5   r   N)r   r   r   r   r   r2   rH   rN   r^   rc   rh   r   r   r   r   s   @r1   rn  rn  V  s5     0dDT"8
8
<
   r5   rn  zipfianz	A Zipfianc                   R   a  ] tR tRt o RtR tR tR tR tR t	R t
RR
 ltRtV tR	# )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                @    \        R R^ \        P                  3R4      .# r[  r,   r/   s   &r1   r2   dlaplace_gen._shape_info  r]  r5   c                h    \        VR ,          4      \        V) \        V4      ,          4      ,          # rw   )r   r   abs)r0   rV   rM   s   &&&r1   r^   dlaplace_gen._pmf  s$    AcE{S!c!f---r5   c                \    \        V4      pR  pR p\        P                  ! V^ 8  W23WE4      # )c                 d    R \        V) V ,          4      \        V4      ^,           ,          ,
          # r   r)  rV   rM   s   &&r1   r   dlaplace_gen._cdf.<locals>.f1  s$    aR!VA
333r5   c                 `    \        W^,           ,          4      \        V4      ^,           ,          # rQ   r)  r  s   &&r1   f2dlaplace_gen._cdf.<locals>.f2  s     qE{#s1vz22r5   )r   r6  r7  )r0   rU   rM   rV   r   r  s   &&&   r1   rc   dlaplace_gen._cdf  s0    !H	4	3 qAvvr66r5   c           
     z   ^\        V4      ,           p\        \        P                  ! VR^\        V) 4      ,           ,          8  \	        W,          4      V,          ^,
          \	        ^V,
          V,          4      ) V,          4      4      pV^,
          p\        P                  ! V P                  WR4      V8  WT4      # )r*   r   )r   r   r-   rD  r   rc   )r0   rr   rM   constrY  r  s   &&&   r1   rs   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                
   \        V4      pR V,          VR,
          ^,          ,          pR V,          V^,          RV,          ,           R,           ,          VR,
          ^,          ,          pRVRWC^,          ,          R,
          3# )rw   r   g      $@r  r*  r)  )r0   rM   ear  r  s   &&   r1   r   dlaplace_gen._stats  se    VeRUQJeRU3r6\"_%B
23CQJO++r5   c                f    V\        V4      ,          \        \        VR ,          4      4      ,
          # r  )r   r   r   rb  s   &&r1   r   dlaplace_gen._entropy  s"    47{Sae---r5   Nc                    \         P                  ! \         P                  ! V4      ) 4      ) pVP                  WBR 7      pVP                  WBR 7      pWV,
          # r?  )r-   r   r=   rA  )r0   rM   r@   rA   probOfSuccessrU   ys   &&&&   r1   rB   dlaplace_gen._rvs   sL      2::a=.11""="<""="<ur5   r   r   )r   r   r   r   r   r2   r^   rc   rs   r   r   rB   r   r   r   s   @r1   r  r    s3     ,E.	7?,. r5   r  dlaplacezA discrete Laplacianc                   l   a  ] tR tRt o RtR tR tRRRR/R ltR	 tR
 t	R t
R tR tV 3R ltRtV tR# )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                    . # rK   r   r/   s   &r1   r2   poisson_binom_gen._shape_infoH  s	     	r5   c                    \         P                  ! V^ R7      p^ V8*  V^8*  ,          p\         P                  ! V^ R7      # )r   r  )r-   stackr:   )r0   argsr'   condss   &*  r1   rH   poisson_binom_gen._argcheckM  s5    HHT"aAF#vve!$$r5   r@   NrA   c               F   \         P                  ! VRR7      pVf   VP                  M1\         P                  ! V4      '       d   V^3M\	        V4      R,           p\         P
                  ! VP                  V4      p\        P                  WAVR7      P                  RR7      # )r*   r  r   rQ   )	r-   r  shapeisscalartuplebroadcast_shapesr   rB   r  )r0   r@   rA   r  r'   s   &$$* r1   rB   poisson_binom_gen._rvsR  s|    HHT#  <[[..q	E$K$4F 	""177D1~~a~FJJPRJSSr5   c                    ^ \        V4      3# rE   )len)r0   r  s   &*r1   rN   poisson_binom_gen._get_support\  s    #d)|r5   c                    \         P                  ! V4      P                  \         P                  4      p\         P                  ! V.VO5!  vr\         P
                  ! V\         P                  RR7      p\        W4      # r*   )r9   r  r  )r-   
atleast_1dr   rE  r   r  re  r   r0   rV   r  r'   s   &&* r1   r^   poisson_binom_gen._pmf_  W    MM!##BHH-&&q040HHT"5!!''r5   c                    \         P                  ! V4      P                  \         P                  4      p\         P                  ! V.VO5!  vr\         P
                  ! V\         P                  RR7      p\        W4      # r  )r-   r  r   rE  r   r  re  r   r  s   &&* r1   rc   poisson_binom_gen._cdfe  r  r5   c                    \         P                  ! V^ R7      p\         P                  ! V^ R7      p\         P                  ! V^V,
          ,          ^ R7      pWERR3# )r   r  N)r-   r  r  )r0   r  kwdsr'   r"  r}   s   &*,   r1   r   poisson_binom_gen._statsk  sG    HHT"vvaa ffQ!A#YQ'4&&r5   c                     \        V .VO5/ VB # rK   )poisson_binomial_frozen)r0   r  r  s   &*,r1   __call__poisson_binom_gen.__call__q  s    &t;d;d;;r5   c                >   < V ^8  d   Qh/ S[ ;R&   S[ ;R&   S[ ;R&   # )r)  _parse_args_rvs_parse_args_stats_parse_argsr   )formatr   s   "r1   __annotate__poisson_binom_gen.__annotate__  s2     T  U V "!W X Y r5   r   )r   r   r   r   r   r2   rH   rB   rN   r^   rc   r   r  __annotate_func__r   r   r   s   @r1   r  r    sL     'Z
%
Tt T$ T(('<o  r5   r  poisson_binomzA Poisson binomialr'   )r   rd  shapesc                 L    \        \        P                  ! VR^ 4      4      VRV3# r*   r   r  r  r-   moveaxis)r0   r'   locr@   s   &&&&r1   r  r  }  s#    QA&'c477r5   c                 L    \        \        P                  ! VR^ 4      4      VRV3# r  r  )r0   r'   r  r{   s   &&&&r1   r  r    s#    QA&'c7::r5   c                 J    \        \        P                  ! VR^ 4      4      VR3# r  r  )r0   r'   r  s   &&&r1   r  r    s!    QA&'c11r5   c                   0   a  ] tR tRt o R tRR ltRtV tR# )r  i  c                   W n         W0n        VP                  ! R/ VP                  4       B V n        \
        P                  \        \        4      V P                  n        \        P                  \        \        4      V P                  n	        \        P                  \        \        4      V P                  n
        V P                  P                  ! V/ VB w  p pV P                  P                  ! V!  w  V n        V n        R # )Nr   )r  r  	__class___updated_ctor_paramdistr  __get___pb_obj_pb_clsr  r  rN   rM   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                    V P                   P                  ! V P                  / V P                  B w  rgpV P                   P                  ! WP                  WrW43/ VB # rK   )r  r  r  r  expect)	r0   funclbubconditionalr  rM   r  scales	   &&&&&,   r1   r  poisson_binomial_frozen.expect  sK    		--tyyFDIIF yyii"RTRRr5   )rM   r  r   r  r  )NNNF)r   r   r   r   r  r  r   r   r   s   @r1   r  r    s     9S Sr5   r  c                   F   a  ] tR tRt o RtR tR
R ltR tR tR t	R	t
V tR# )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.

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

%(after_notes)s

References
----------
.. [1] Skellam, J. G. "The Frequency Distribution of the Difference
       Between Two Poisson Variates Belonging to Different Populations."
       *Journal of the Royal Statistical Society* 109, no. 3 (1946): 296-296.
       :doi:`10.2307/2981372`
.. [2] "Skellam distribution", Wikipedia,
       https://en.wikipedia.org/wiki/Skellam_distribution

%(example)s

c                z    \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )r  Fr  r   r,   r/   s   &r1   r2   skellam_gen._shape_info  s:    5%!RVVnE5%!RVVnEG 	Gr5   Nc                T    TpVP                  W4      VP                  W%4      ,
          # rK   r  )r0   r  r  r@   rA   r&   s   &&&&& r1   rB   skellam_gen._rvs  s-    $$S,$$S,- 	.r5   c                   \         P                  ! R R7      ;_uu_ 4        \         P                  ! V^ 8  \        P                  ! ^V,          ^^V,
          ,          ^V,          4      ^,          \        P                  ! ^V,          ^^V,           ,          ^V,          4      ^,          4      pRRR4       V#   + '       g   i     X# ; ir  )r-   r   rD  r[   	_ncx2_pdfr0   rU   r  r  pxs   &&&& r1   r^   skellam_gen._pmf  s    [[h''!a%--#q!A#w#>q@--#q!A#w#>q@BB (
 	 ('
 	s   BB88C		c                   \        V4      p\        P                  ! R R7      ;_uu_ 4        \        P                  ! V^ 8  \        P
                  ! ^V,          RV,          ^V,          4      \        P                  ! ^V,          ^V^,           ,          ^V,          4      4      pRRR4       V#   + '       g   i     X# ; i)r   r  N)r   r-   r   rD  r   chndtrr[   _ncx2_sfr  s   &&&& r1   rc   skellam_gen._cdf  s    !H[[h''!a%!..31ae<,,qua1gqu=?B ( 		 (' 	s   A9B..B?	c                n    W,
          pW,           pV\        V^,          4      ,          p^V,          pW4WV3# )   r+  )r0   r  r  r"  r}   r~   r   s   &&&    r1   r   skellam_gen._stats  s6    yiD#N"W"  r5   r   r   )r   r   r   r   r   r2   rB   r^   rc   r   r   r   r   s   @r1   r  r    s*     #HG.
! !r5   r  skellamz	A Skellamc                   ^   a  ] tR tRt o RtR tRR ltR tR tR t	R	 t
R
 tR tR tRtV tR# )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                @    \        R R^ \        P                  3R4      .# )alphaFr   r,   r/   s   &r1   r2   yulesimon_gen._shape_info  s    7EArvv;GHHr5   Nc           	         VP                  V4      pVP                  V4      p\        V) \        \        V) V,          4      ) 4      ,          4      pV# rK   )standard_exponentialr   r   r   )r0   r  r@   rA   E1E2anss   &&&&   r1   rB   yulesimon_gen._rvs  sK    ..t4..t4B3RC%K 00112
r5   c                J    V\         P                  ! W^,           4      ,          # rQ   r   r   r0   rU   r  s   &&&r1   r^   yulesimon_gen._pmf  s    w||Aqy111r5   c                    V^ 8  # rE   r   )r0   r  s   &&r1   rH   yulesimon_gen._argcheck  s    	r5   c                \    \        V4      \        P                  ! W^,           4      ,           # rQ   r   r   r	   r  s   &&&r1   rX   yulesimon_gen._logpmf  s    5zGNN1ai888r5   c                X    ^V\         P                  ! W^,           4      ,          ,
          # rQ   r  r  s   &&&r1   rc   yulesimon_gen._cdf"  s    1w||Aqy1111r5   c                J    V\         P                  ! W^,           4      ,          # rQ   r  r  s   &&&r1   rh   yulesimon_gen._sf%  s    7<<19---r5   c                \    \        V4      \        P                  ! W^,           4      ,           # rQ   r  r  s   &&&r1   rT  yulesimon_gen._logsf(  s    1vq!)444r5   c                   \         P                  ! V^8*  \         P                  W^,
          ,          4      p\         P                  ! V^8  V^,          VR,
          V^,
          ^,          ,          ,          \         P                  4      p\         P                  ! V^8*  \         P                  V4      p\         P                  ! V^8  \	        V^,
          4      V^,           ^,          ,          W^,
          ,          ,          \         P                  4      p\         P                  ! V^8*  \         P                  V4      p\         P                  ! V^8  V^,           ^V^,          ,          ^1V,          ,
          ^,
          W^,
          ,          V^,
          ,          ,          ,           \         P                  4      p\         P                  ! V^8*  \         P                  V4      pW#WE3# r  )r-   rD  r.   nanr   )r0   r  r|   r  r~   r   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   r   )r   r   r   r   r   r2   rB   r^   rH   rX   rc   rh   rT  r   r   r   r   s   @r1   r  r    s>      BI292.5 r5   r  	yulesimon)r   rM   c                   X   a  ] tR tRt o RtRtRtR tR tR t	RR lt
R tRR	 ltR
tV tR# )_nchypergeom_geni@  zA noncentral hypergeometric discrete random variable.

For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.

Nc           	         \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# )rh  TFr&   ri  oddsr(   r   r,   r/   s   &r1   r2   _nchypergeom_gen._shape_infoJ  sf    3q"&&k=A3q"&&k=A3q"&&k=A651bff+~FH 	Hr5   c                    YTr%pW5,
          p\         P                  ! ^ W&,
          4      p\         P                  ! W%4      pWx3# rE   rp  )	r0   rh  r&   ri  r$  rM  rL  x_minx_maxs	   &&&&&    r1   rN   _nchypergeom_gen._get_supportP  s:    qV

1af%

1!|r5   c                   \         P                  ! V4      \         P                  ! V4      r!\         P                  ! V4      \         P                  ! V4      rC\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8  ,          p\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8  ,          p\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8  ,          pV^ 8  pW18*  p	W!8*  p
WV,          V,          V,          V	,          V
,          # rE   )r-   r=   isnanr   r>   )r0   rh  r&   ri  r$  cond1cond2cond3cond4cond5cond6s   &&&&&      r1   rH   _nchypergeom_gen._argcheckW  s    zz!}bjjm1**Q-D!14((1+!((3-1"45a@((1+!((3-1"45a@((1+!((3-1"45a@q}u$u,u4u<<r5   c           	     8   a  \         V 3R  l4       pV! WW4WVR7      # )c                   < \         P                  ! V 4      \         P                  ! V4      ,          \         P                  ! V4      ,          '       d&   \         P                  ! V\         P                  4      # \         P                  ! V4      p\        4       p\        VS
P                  4      pV! W!WWe4      p	V	P                  V4      p	V	# rK   )	r-   r+  fullr  prodr!   getattrrvs_namereshape)rh  r&   ri  r$  r@   rA   lengthurnrv_genr  r0   s   &&&&&&    r1   r  $_nchypergeom_gen._rvs.<locals>._rvs1d  s    xx{RXXa[(288A;66wwtRVV,,WWT]F#%CS$--0Fq=C++d#CJr5   r   r  )r0   rh  r&   ri  r$  r@   rA   r  s   f&&&&&& r1   rB   _nchypergeom_gen._rvsb  s&    	#	 
$	 Q1IIr5   c                   a  \         P                  ! WW4V4      w  rr4pVP                  ^ 8X  d   \         P                  ! V4      # \         P                  V 3R l4       pV! WW4V4      # )r   c                 F  < \         P                  ! V 4      \         P                  ! V4      ,          \         P                  ! V4      ,          \         P                  ! V4      ,          '       d   \         P                  # SP                  W2WR 4      pVP	                  V 4      # g-q=)r-   r+  r  r  probability)rU   rh  r&   ri  r$  r;  r0   s   &&&&& r1   _pmf1$_nchypergeom_gen._pmf.<locals>._pmf1w  sb    xx{RXXa[(288A;6!DDvv))A!51C??1%%r5   )r-   r   r@   
empty_likerS  )r0   rU   rh  r&   ri  r$  rC  s   f&&&&& r1   r^   _nchypergeom_gen._pmfq  s_    ..qQ4@aD66Q;==##		& 
	& Q1&&r5   c                z   a  \         P                  V 3R  l4       pRV9   g   RV9   d
   V! WW44      MRw  rxRRrWxW3# )c                 .  < \         P                  ! V 4      \         P                  ! V4      ,          \         P                  ! V4      ,          '       d!   \         P                  \         P                  3# SP                  W!WR 4      pVP	                  4       # rA  )r-   r+  r  r  r{   )rh  r&   ri  r$  r;  r0   s   &&&& r1   	_moments1*_nchypergeom_gen._stats.<locals>._moments1  s[    xx{RXXa[(288A;66vvrvv~%))A!51C;;= r5   r  vNr   )r-   rS  )r0   rh  r&   ri  r$  r{   rI  r  rK  rv   rV   s   f&&&&&     r1   r   _nchypergeom_gen._stats  sL    		! 
	! .1G^sg~	!(! 	T1Qzr5   r   r   r   )r   r   r   r   r   r8  r  r2   rN   rH   rB   r^   r   r   r   r   s   @r1   r"  r"  @  s;      HDH	=J' r5   r"  c                   "    ] tR tRtRtRt]tRtR# )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   r8  r   r  r   r   r5   r1   rN  rN    s    GR H%Dr5   rN  nchypergeom_fisherz$A Fisher's noncentral hypergeometricc                   "    ] tR tRtRtRt]tRtR# )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   r8  r    r  r   r   r5   r1   rR  rR    s    GR H'Dr5   rR  nchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)r   N)r   r   rE   )l	functoolsr   typesr   scipyr   scipy.specialr   r   r	   r
   rR   scipy.special._ufuncs_ufuncsr[   scipy.special._spfun_statsr   r   scipy._lib._utilr   scipy._external.array_api_extra	_externalarray_api_extrar6  scipy.interpolater   numpyr   r   r   r   r   r   r   r   r   r   r-   _distn_infrastructurer   r   r   r   r   r   
_biasedurnr   r    r!   r#   r   r   r   r   r   r   r  r  r9  r;  rc  rf  r  r  r  r  r  r  r  r  r  r   r7  r9  rT  rY  r_  rn  r  r  r.   r  r  r  r  r  r  r  r  r  r  r  r  r  r   r"  rN  rP  rR  rT  listglobalscopyitemspairs_distn_names_distn_gen_names__all__r   r5   r1   <module>rl     s  
    C C # # M ) - - & M M M 8 8, ,
Z Zz 	w>#I >#B AK0	QK Qh {+	r
 r
j 
	"Z[ Zz .
M7{ M7` !&=9WK Wt {+	Z[ Zz .
= =@ 
ah	A?+ ?D 9{
;C0 C0L 
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?{ ?D !&84j + j Z 	K
@M; M` 266''2HJX< X<v "AU),.8;2
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)
)r5   