+
    Z°³jë ã                   óø  € ^ 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]TP³                  ]W]X4      ]SnT        ]UP³                  ]W]X4      ]SnU        ]VP³                  ]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       PÍ                  4       PÏ                  4       4      th](! ]h]'4      w  titj]i]j,           tkR# )jé    )Úpartial©Ú
MethodType)Úspecial)ÚentrÚ	logsumexpÚbetalnÚgammalnN)Ú_poisson_binom_pmfÚ_poisson_binom_cdf)Úrng_integers)Úinterp1d)
ÚfloorÚceilÚlogÚexpÚsqrtÚlog1pÚexpm1ÚtanhÚcoshÚsinh)Úrv_discreteÚget_distribution_namesÚ_vectorize_rvs_over_shapesÚ
_ShapeInfoÚ_isintegralÚrv_discrete_frozen)Ú_PyFishersNCHypergeometricÚ_PyWalleniusNCHypergeometricÚ_PyStochasticLib3c                   ón   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# )Ú	binom_genaÚ  
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                óZ   € \        R R^ \        P                  3R4      \        RRRR4      .# ©ÚnTFÚp©TF©r   é   ©TT©r   ÚnpÚinf©Úselfs   &Ú]/home/luca/screen-checking/.venv/lib/python3.14/site-packages/scipy/stats/_discrete_distns.pyÚ_shape_infoÚbinom_gen._shape_infoA   ó0   € Ü˜3  q¬"¯&©& k°=ÓAÜ˜3  v¨|Ó<ð>ð 	>ó    Nc                óÖ   € \         P                  ! V\         P                  ! V4      8H  4      '       g   \        R 4      hVP	                  \         P
                  ! V\        R7      W#4      # ©z`n` must be integral.©Údtype)r-   Úallr   Ú
ValueErrorÚbinomialÚasarrayÚint©r0   r&   r'   ÚsizeÚrandom_states   &&&&&r1   Ú_rvsÚbinom_gen._rvsE   sH   € Ü�vŠv�aœ2Ÿ8š8 A›;Ñ&×'Ò'ÜÐ4Ó5Ð5Ø×$Ñ$¤R§Z¢Z°¼Ô%=¸qÓGÐGr5   c                óT   € V^ 8¬  \        V4      ,          V^ 8¬  ,          V^8*  ,          # ©r   ©r   ©r0   r&   r'   s   &&&r1   Ú	_argcheckÚbinom_gen._argcheckJ   s'   € Ø�Q‘œ+ a›.Õ(¨A°©FÕ3°q¸A±vÕ>Ð>r5   c                ó   € V P                   V3# ©N©ÚarG   s   &&&r1   Ú_get_supportÚbinom_gen._get_supportM   s   € Ø�v‰v�qˆyÐr5   c                ó$  € \        V4      p\        V^,           4      \        V^,           4      \        W$,
          ^,           4      ,           ,
          pV\        P                  ! WC4      ,           \        P                  ! W$,
          V) 4      ,           # ©r*   )r   Úgamlnr   ÚxlogyÚxlog1py)r0   Úxr&   r'   ÚkÚcombilns   &&&&  r1   Ú_logpmfÚbinom_gen._logpmfP   s\   € Ü�!‹HˆÜ˜˜1�“:¤ q¨¥s£¬e°AµC¸µE«lÕ!:Õ;ˆØœŸš qÓ,Õ,¬w¯ª¸q½sÀQÀBÓ/GÕGÐGr5   c                ó0   € \         P                  ! WV4      # rK   )ÚscuÚ
_binom_pmf©r0   rU   r&   r'   s   &&&&r1   Ú_pmfÚbinom_gen._pmfU   s   € ä�~Š~˜a AÓ&Ð&r5   c                óF   € \        V4      p\        P                  ! WBV4      # rK   )r   r[   Ú
_binom_cdf©r0   rU   r&   r'   rV   s   &&&& r1   Ú_cdfÚbinom_gen._cdfY   ó   € Ü�!‹HˆÜ�~Š~˜a AÓ&Ð&r5   c                óF   € \        V4      p\        P                  ! WBV4      # rK   )r   r[   Ú	_binom_sfrb   s   &&&& r1   Ú_sfÚbinom_gen._sf]   s   € Ü�!‹HˆÜ�}Š}˜Q 1Ó%Ð%r5   c                ó0   € \         P                  ! WV4      # rK   )r[   Ú
_binom_isfr]   s   &&&&r1   Ú_isfÚbinom_gen._isfa   ó   € Ü�~Š~˜a AÓ&Ð&r5   c                ó0   € \         P                  ! WV4      # rK   )r[   Ú
_binom_ppf©r0   Úqr&   r'   s   &&&&r1   Ú_ppfÚbinom_gen._ppfd   rn   r5   c                óè  € 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# )NÚsç       @rV   ç      @)r-   Úsquarer   Ú
reciprocal)r0   r&   r'   ÚmomentsÚmuÚvarÚg1Úg2ÚpqÚnpq_sqrtÚt1Út2Únpqs   &&&&         r1   Ú_statsÚbinom_gen._statsg   s¸   € Ø�UˆØ”r—y’y “|Õ#Õ#ˆØ�tˆBØ�'Œ>Ø”R—Y’Y˜q“\Õ!ˆBÜ—w’w˜q�v“ˆHÜ—’˜xÓ(ˆBØ˜•'˜XÕ%ˆBØ•ˆBØ�'Œ>Ø”R—Y’Y˜q“\Õ!ˆBØ•&ˆCÜ—’˜sÓ#ˆBØ�Q•ˆBØ•ˆBØ˜ˆÐr5   © ©NN©Úmv)Ú__name__Ú
__module__Ú__qualname__Ú__firstlineno__Ú__doc__r2   rB   rH   rN   rX   r^   rc   rh   rl   rs   r…   Ú__static_attributes__Ú__classdictcell__©Ú__classdict__s   @r1   r#   r#      sI   ø‡ € ñ"òF>ôHò
?òòHò
'ò'ò&ò'ò'÷ò r5   r#   Úbinom)Únamec                   óp   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# )Úbernoulli_genaç  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                ó    € \        R RRR4      .# ©r'   Fr)   r+   ©r   r/   s   &r1   r2   Úbernoulli_gen._shape_info–   ó   € Ü˜3  v¨|Ó<Ð=Ð=r5   Nc                ó4   € \         P                  V ^WVR7      # )r*   ©r@   rA   )r#   rB   ©r0   r'   r@   rA   s   &&&&r1   rB   Úbernoulli_gen._rvs™   s   € Ü�~‰~˜d A qÀ,ˆ~ÓOÐOr5   c                ó    € V^ 8¬  V^8*  ,          # rE   r‡   ©r0   r'   s   &&r1   rH   Úbernoulli_gen._argcheckœ   s   € Ø�Q‘˜1 ™6Õ"Ð"r5   c                ó2   € V P                   V P                  3# rK   )rM   Úbr¢   s   &&r1   rN   Úbernoulli_gen._get_supportŸ   s   € à�v‰v�t—v‘vˆ~Ðr5   c                ó0   € \         P                  V^V4      # rQ   )r”   rX   ©r0   rU   r'   s   &&&r1   rX   Úbernoulli_gen._logpmf£   s   € Ü�}‰}˜Q  1Ó%Ð%r5   c                ó0   € \         P                  V^V4      # rQ   )r”   r^   r¨   s   &&&r1   r^   Úbernoulli_gen._pmf¦   s   € ô �z‰z˜!˜Q Ó"Ð"r5   c                ó0   € \         P                  V^V4      # rQ   )r”   rc   r¨   s   &&&r1   rc   Úbernoulli_gen._cdf«   ó   € Ü�z‰z˜!˜Q Ó"Ð"r5   c                ó0   € \         P                  V^V4      # rQ   )r”   rh   r¨   s   &&&r1   rh   Úbernoulli_gen._sf®   s   € Ü�y‰y˜˜A˜qÓ!Ð!r5   c                ó0   € \         P                  V^V4      # rQ   )r”   rl   r¨   s   &&&r1   rl   Úbernoulli_gen._isf±   r®   r5   c                ó0   € \         P                  V^V4      # rQ   )r”   rs   )r0   rr   r'   s   &&&r1   rs   Úbernoulli_gen._ppf´   r®   r5   c                ó.   € \         P                  ^V4      # rQ   )r”   r…   r¢   s   &&r1   r…   Úbernoulli_gen._stats·   s   € Ü�|‰|˜A˜qÓ!Ð!r5   c                óF   € \        V4      \        ^V,
          4      ,           # rQ   )r   r¢   s   &&r1   Ú_entropyÚbernoulli_gen._entropyº   s   € Ü�A‹wœ˜a �c›Õ"Ð"r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   rN   rX   r^   rc   rh   rl   rs   r…   r¸   r�   r‘   r’   s   @r1   r—   r—   }   sL   ø‡ € ñò0>ôPò#òò&ò#ò
#ò"ò#ò#ò"÷#ð #r5   r—   Ú	bernoulli)r¥   r•   c                   óV   a € ] tR t^Át 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# )Úbetabinom_gena¾  
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                ó´   € \        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   Úbetabinom_gen._shape_infoå   óP   € Ü˜3  q¬"¯&©& k°=ÓAÜ˜3 ¨¬2¯6©6 {°NÓCÜ˜3 ¨¬2¯6©6 {°NÓCðEð 	Er5   Nc                óú   € 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   )Úbetar-   r:   r   r;   r<   r=   r>   ©r0   r&   rM   r¥   r@   rA   r'   s   &&&&&& r1   rB   Úbetabinom_gen._rvsê   sZ   € Ø×Ñ˜a DÓ)ˆÜ�vŠv�aœ2Ÿ8š8 A›;Ñ&×'Ò'ÜÐ4Ó5Ð5Ø×$Ñ$¤R§Z¢Z°¼Ô%=¸qÓGÐGr5   c                ó
   € ^ V3# rE   r‡   ©r0   r&   rM   r¥   s   &&&&r1   rN   Úbetabinom_gen._get_supportð   ó   € Ø�!ˆtˆr5   c                óT   € V^ 8¬  \        V4      ,          V^ 8„  ,          V^ 8„  ,          # rE   rF   rÇ   s   &&&&r1   rH   Úbetabinom_gen._argcheckó   ó'   € Ø�Q‘œ+ a›.Õ(¨A°©EÕ2°a¸!±eÕ<Ð<r5   c                óô   € \        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   Úbetabinom_gen._logpmfö   sS   € Ü�!‹HˆÜ�q˜1•u“:�+¤ q¥u¨q¥y°!°aµ%Ó 8Õ8ˆØœ ¥ q¥u¨q¥yÓ1Õ1´F¸1³LÕ@Ð@r5   c                ó8   € \        V P                  WW44      4      # rK   ©r   rX   ©r0   rU   r&   rM   r¥   s   &&&&&r1   r^   Úbetabinom_gen._pmfû   ó   € Ü�4—<‘<  aÓ+Ó,Ð,r5   c                óL  € W"V,           ,          p^V,
          pW,          pWV,           V,           ,          V,          V,          W#,           ^,           ,          pRRr©RV9   d`   R\        V4      ,          p	W’V,           ^V,          ,           W2,
          ,          ,          p	W’V,           ^,           W#,           ,          ,          p	RV9   EdY   W#,           P                  VP                  4      p
W¢V,           ^,
          ^V,          ,           ,          p
V
^V,          V,          V^,
          ,          ,          p
V
^V^,          ,          ,          p
V
^V,          V,          V,          ^V,
          ,          ,          p
V
^V,          V,          V^,          ,          ,          p
W¢V,           ^,          ^V,           V,           ,          ,          p
W¡V,          V,          W#,           ^,           ,          W#,           ^,           ,          W#,           V,           ,          ,          p
V
^,          p
WxWš3# )r*   Nrv   ç      ð?rV   )r   Úastyper9   )r0   r&   rM   r¥   r{   Úe_pÚe_qr|   r}   r~   r   s   &&&&&      r1   r…   Úbetabinom_gen._statsþ   sŠ  € Ø�q•5�kˆØ�#�gˆØ�WˆØ�q•5˜1•9�o Õ# cÕ)¨Q­U°Q­YÕ7ˆØ�tˆBØ�'Œ>Ø”t˜C“y•ˆBØ�q•5˜1˜q�5•= Q¥UÕ+Õ+ˆBØ�q•5˜1•9 ¥Õ'Õ'ˆBØ�'�>Ø•%—‘ §	¡	Ó*ˆBØ�q•5˜1•9˜q 1�uÕ$Õ%ˆBØ�!�a•%˜!•)˜q 1�uÕ%Õ%ˆBØ�!�a˜1•f•*ÕˆBØ�!�c•'˜A•+ •/ Q¨¥UÕ+Õ+ˆBØ�"�s•(˜S•. 1¨¥6Õ)Õ)ˆBØ�q•5˜Q•, ! a¥%¨!¥)Õ,Õ,ˆBØ�q•5˜1•9 ¥¨¥	Õ*¨a­e°a­iÕ8¸A½EÀA½IÕFÕGˆBØ�!�GˆBØ˜ˆÐr5   r‡   rˆ   r‰   )r‹   rŒ   r�   rŽ   r�   r2   rB   rN   rH   rX   r^   r…   r�   r‘   r’   s   @r1   r¼   r¼   Á   s6   ø‡ € ñ"òFEô
Hòò=òAò
-÷ò r5   r¼   Ú	betabinomc                   ój   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# )Ú
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                óZ   € \        R R^ \        P                  3R4      \        RRRR4      .# r%   r,   r/   s   &r1   r2   Únbinom_gen._shape_infoS  r4   r5   Nc                ó&   € VP                  WV4      # rK   )Únegative_binomialr?   s   &&&&&r1   rB   Únbinom_gen._rvsW  s   € Ø×-Ñ-¨a°DÓ9Ð9r5   c                ó4   € V^ 8„  V^ 8„  ,          V^8*  ,          # rE   r‡   rG   s   &&&r1   rH   Únbinom_gen._argcheckZ  s   € Ø�A‘˜!˜a™%Õ  A¨¡FÕ+Ð+r5   c                ó0   € \         P                  ! WV4      # rK   )r[   Ú_nbinom_pmfr]   s   &&&&r1   r^   Únbinom_gen._pmf]  s   € ä�Š˜q QÓ'Ð'r5   c                óÚ   € \        W!,           4      \        V^,           4      ,
          \        V4      ,
          pWB\        V4      ,          ,           \        P                  ! W) 4      ,           # rQ   )rR   r   r   rT   )r0   rU   r&   r'   Úcoeffs   &&&& r1   rX   Únbinom_gen._logpmfa  sD   € Ü�a•c“
œU 1 Q¥3›ZÕ'¬%°«(Õ2ˆØœ˜Q›•xÕ¤'§/¢/°!°RÓ"8Õ8Ð8r5   c                óF   € \        V4      p\        P                  ! WBV4      # rK   )r   r[   Ú_nbinom_cdfrb   s   &&&& r1   rc   Únbinom_gen._cdfe  s   € Ü�!‹HˆÜ�Š˜q QÓ'Ð'r5   c                ó|  € \        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)ç      à?c                 óx   € \         P                  ! \        P                  ! V ^,           V^V,
          4      ) 4      # rQ   )r-   r   r   Úbetainc)rV   r&   r'   s   &&&r1   Úf1Únbinom_gen._logcdf.<locals>.f1n  s)   € Ü—8’8œWŸ_š_¨Q°­U°A°q¸1µuÓ=Ð=Ó>Ð>r5   Úignore)ÚdivideN)r   r-   Úbroadcast_arraysrc   Úerrstater   )	r0   rU   r&   r'   rV   ÚcdfÚcondrò   Úlogcdfs	   &&&&     r1   Ú_logcdfÚnbinom_gen._logcdfi  s�   € Ü�!‹HˆÜ×%Ò% a¨AÓ.‰ˆˆaØ�i‰i˜˜aÓ ˆØ�S‰yˆò	?ð ˆÜ�[Š[ ×)Ö)Ù˜a�g q¥w°µÓ8ˆF‰LÜŸFšF 3 u¥:Ó.ˆF�5‰M÷ *ð ˆ÷ *Ö)ð ˆús   Á!?B*Â*B;	c                óF   € \        V4      p\        P                  ! WBV4      # rK   )r   r[   Ú
_nbinom_sfrb   s   &&&& r1   rh   Únbinom_gen._sfx  re   r5   c                ó¬   € \         P                  ! R R7      ;_uu_ 4        \        P                  ! WV4      uuRRR4       #   + '       g   i     R# ; i©rô   ©ÚoverN)r-   r÷   r[   Ú_nbinom_isfr]   s   &&&&r1   rl   Únbinom_gen._isf|  ó.   € Ü�[Š[˜h×'Ö'Ü—?’? 1¨Ó+÷ (×'×'Ó'úó    AÁA	c                ó¬   € \         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[   Ú_nbinom_ppfrq   s   &&&&r1   rs   Únbinom_gen._ppf€  r  r  c                ó®   € \         P                  ! W4      \         P                  ! W4      \         P                  ! W4      \         P                  ! W4      3# rK   )r[   Ú_nbinom_meanÚ_nbinom_varianceÚ_nbinom_skewnessÚ_nbinom_kurtosis_excessrG   s   &&&r1   r…   Únbinom_gen._stats„  sD   € ä×Ò˜QÓ"Ü× Ò  Ó&Ü× Ò  Ó&Ü×'Ò'¨Ó-ð	
ð 	
r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   r^   rX   rc   rû   rh   rl   rs   r…   r�   r‘   r’   s   @r1   rÝ   rÝ     sG   ø‡ € ñ9òt>ô:ò,ò(ò9ò(òò'ò,ò,÷
ð 
r5   rÝ   Únbinomc                   óP   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# )Úbetanbinom_geni�  aó  
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                ó´   € \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# r¾   r,   r/   s   &r1   r2   Úbetanbinom_gen._shape_infoµ  rÁ   r5   Nc                óJ   € VP                  W#V4      pVP                  WV4      # rK   )rÃ   rá   rÄ   s   &&&&&& r1   rB   Úbetanbinom_gen._rvsº  s'   € Ø×Ñ˜a DÓ)ˆØ×-Ñ-¨a°DÓ9Ð9r5   c                óT   € V^ 8¬  \        V4      ,          V^ 8„  ,          V^ 8„  ,          # rE   rF   rÇ   s   &&&&r1   rH   Úbetanbinom_gen._argcheck¾  rÌ   r5   c                óÞ   € \        V4      p\        P                  ! W%,           4      ) \        W%^,           4      ,
          pV\        W2,           WE,           4      ,           \        W44      ,
          # rQ   )r   r-   r   r	   rÎ   s   &&&&&  r1   rX   Úbetanbinom_gen._logpmfÁ  sI   € Ü�!‹HˆÜ—6’6˜!�%“=�.¤6¨!°­UÓ#3Õ3ˆØœ ¥ q¥uÓ-Õ-´°q³Õ<Ð<r5   c                ó8   € \        V P                  WW44      4      # rK   rÑ   rÒ   s   &&&&&r1   r^   Úbetanbinom_gen._pmfÆ  rÔ   r5   c                ó¬  € 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r˜R 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	WgW‰3# )c                 ó.   € W,          VR ,
          ,          # ©rÖ   r‡   ©r&   rM   r¥   s   &&&r1   ÚmeanÚ#betanbinom_gen._stats.<locals>.meanÌ  s   € Ø•5˜A �FÕ#Ð#r5   ©Ú
fill_valuec                 ó¨   € W,          W,           R ,
          ,          W,           R ,
          ,          VR,
          VR ,
          R,          ,          ,          # )rÖ   rw   r‡   r!  s   &&&r1   r}   Ú"betanbinom_gen._stats.<locals>.varÏ  s;   € Ø•E˜Q�U R�ZÕ(¨A­E°B­JÕ7Ø˜B� 1 r¥6¨B¥,Õ.õ0ð 1r5   Nc                 ó  € ^V ,          V,           R,
          ^V,          V,           R,
          ,          VR,
          ,          \        W,          W,           R,
          ,          W!,           R,
          ,          VR,
          ,          4      ,          # )é   rÖ   ç      @rw   ©r   r!  s   &&&r1   ÚskewÚ#betanbinom_gen._stats.<locals>.skewÔ  sf   € Ø˜•U˜Q•Y •^¨¨A­°­	°B­Õ7Ø˜2•võÜ!% a¥e¨q­u°r­zÕ&:¸a½eÀb½jÕ&IØ˜2•võ'ó " õ ð !r5   rv   c                 óT  € 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   rÖ   rx   r*  ç      @g      @r‡   )r&   rM   r¥   ÚtermÚterm_2Údenominators   &&&   r1   ÚkurtosisÚ'betanbinom_gen._stats.<locals>.kurtosisÚ  sH  € Ø˜•FˆDØ˜2•v •l a¨¥e¨a°1°qµ5¸2µ:Õ.>Õ&>Ø˜a "�f�¨Õ)õ'*õ +à˜Q �U�
 q¨2¥v°°BµÕ&6¸!¸b½&Ø˜R�õ:!Ø#$õ:%õ '%Ø')¨Q°­V°a­KÕ'7õ'8õ 9õ9ð ˜Q �V� qÕ(Ø˜b�& A r¥EÕ)¨Q°­V¸¸B½Õ,?À!Õ,CÕCØ˜a "�f r�\Õ)õ*õ+õ	+ˆFð  �F q¨2¥vÕ.°Õ2°QÕ6Ø�e b�jõ*Ø-.­U°R­Zõ9ˆKð •= ;Õ.°Õ3Ð3r5   rV   ©ÚxpxÚapply_wherer-   r.   )r0   r&   rM   r¥   r{   r"  r|   r}   r~   r   r,  r3  s   &&&&&       r1   r…   Úbetanbinom_gen._statsÉ  s»   € ò	$ä�_Š_˜Q ™U Q¨1 I¨tÄÇÁÔGˆò	1ô �oŠo˜a !™e a¨A Y°ÄÇÁÔGˆØ�tˆBò	!ð �'Œ>Ü—’  Q¡¨¨q¨	°4ÄBÇFÁFÔKˆBò	4ð �'Œ>Ü—’  Q¡¨¨q¨	°8ÌÏÉÔOˆBØ˜ˆÐr5   r‡   rˆ   r‰   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   rX   r^   r…   r�   r‘   r’   s   @r1   r  r  �  s/   ø‡ € ñ#òHEô
:ò=ò=ò
-÷!ò !r5   r  Ú
betanbinomc                   ój   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# )Úgeom_genið  aä  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                ó    € \        R RRR4      .# r™   rš   r/   s   &r1   r2   Úgeom_gen._shape_info  rœ   r5   Nc                ó®   € VP                  WR 7      p\        P                  ! VP                  4      P                  p\        P
                  ! V^ 8  WT4      # ©©r@   )Ú	geometricr-   Úiinfor9   ÚmaxÚwhere)r0   r'   r@   rA   ÚresÚmax_ints   &&&&  r1   rB   Úgeom_gen._rvs  sD   € Ø×$Ñ$ QÐ$Ó2ˆô —(’(˜3Ÿ9™9Ó%×)Ñ)ˆÜ�xŠx˜˜a™ Ó.Ð.r5   c                ó    € V^8*  V^ 8„  ,          # rQ   r‡   r¢   s   &&r1   rH   Úgeom_gen._argcheck  s   € Ø�Q‘˜1˜q™5Õ!Ð!r5   c                óZ   € \         P                  ! ^V,
          V^,
          4      V,          # rQ   )r-   Úpower©r0   rV   r'   s   &&&r1   r^   Úgeom_gen._pmf  s    € Ü�xŠx˜˜!�˜Q˜q�SÓ! AÕ%Ð%r5   c                ó`   € \         P                  ! V^,
          V) 4      \        V4      ,           # rQ   )r   rT   r   rL  s   &&&r1   rX   Úgeom_gen._logpmf!  s"   € Ü�Š˜q 1�u q bÓ)¬C°«FÕ2Ð2r5   c                óR   € \        V4      p\        \        V) 4      V,          4      ) # rK   )r   r   r   ©r0   rU   r'   rV   s   &&& r1   rc   Úgeom_gen._cdf$  s#   € Ü�!‹HˆÜ”e˜Q˜B“i •kÓ"Ð"Ð"r5   c                óL   € \         P                  ! V P                  W4      4      # rK   )r-   r   Ú_logsfr¨   s   &&&r1   rh   Úgeom_gen._sf(  s   € Ü�vŠv�d—k‘k !Ó'Ó(Ð(r5   c                ó>   € \        V4      pV\        V) 4      ,          # rK   )r   r   rQ  s   &&& r1   rT  Úgeom_gen._logsf+  s   € Ü�!‹HˆØ”˜�r“�{Ðr5   c                óÖ   € \        \        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'   ÚvalsÚtemps   &&&  r1   rs   Úgeom_gen._ppf/  sS   € Ü”E˜1˜"“I¤ q b£	Õ)Ó*ˆØ�y‰y˜˜a� Ó#ˆÜ�xŠx˜™ t¨a¡xÕ0°$°qµ&¸$Ó?Ð?r5   c                óÌ   € R V,          pR V,
          pW1,          V,          pRV,
          \        V4      ,          p\        P                  ! . ROV4      R V,
          ,          pW$WV3# )rÖ   rw   )r*   iúÿÿÿé   )r   r-   Úpolyval)r0   r'   r|   Úqrr}   r~   r   s   &&     r1   r…   Úgeom_gen._stats4  sT   € Ø��UˆØ��UˆØ�f�q�jˆØ�!�e”t˜B“xÕˆÜ�ZŠZš
 AÓ&¨¨A­Õ.ˆØ˜ˆÐr5   c                ó’   € \         P                  ! V4      ) \         P                  ! V) 4      R V,
          ,          V,          ,
          # r   )r-   r   r   r¢   s   &&r1   r¸   Úgeom_gen._entropy<  s/   € Ü—’�q“	ˆzœBŸHšH a R›L¨C°­EÕ2°QÕ6Õ6Ð6r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   r^   rX   rc   rh   rT  rs   r…   r¸   r�   r‘   r’   s   @r1   r;  r;  ð  sH   ø‡ € ñò@>ô/ò"ò&ò3ò#ò)òò@ò
÷7ð 7r5   r;  ÚgeomzA geometric)rM   r•   Úlongnamec                   óp   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# )Ú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)
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          W4R 7      # r?  )Úhypergeometric)r0   rh  r&   ri  r@   rA   s   &&&&&&r1   rB   Úhypergeom_gen._rvs‘  s   € Ø×*Ñ*¨1°­c°1Ð*Ó@Ð@r5   c                óv   € \         P                  ! W1V,
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r5   c                óÈ   € \         P                  W1V,
          ,
          \        W#4      ^,            pV P                  WAW#4      p\         P                  ! \        V4      ^ R7      # )r*   ©Úaxis)r-   Úr_ÚminÚpmfÚsumr   )r0   rh  r&   ri  rV   rY  s   &&&&  r1   r¸   Úhypergeom_gen._entropy¼  sE   € Ü�E‰E�!˜1•u•+œc !›i¨!�mÐ,ˆØ�x‰x˜˜aÓ#ˆÜ�vŠv”d˜4“j qÔ)Ð)r5   c                ó0   € \         P                  ! WWB4      # rK   )r[   Ú_hypergeom_sfr  s   &&&&&r1   rh   Úhypergeom_gen._sfÁ  s   € Ü× Ò   qÓ,Ð,r5   c                óî  € . p\        \        P                  ! WW44      !   F¼  w  rgr‰VR ,           VR ,           ,          VR ,
          V	R ,
          ,          8  d7   VP                  \	        \        V P                  WgW‰4      4      ) 4      4       Km  \        P                  ! V^,           V	^,           4      p
VP                  \        V P                  W§W‰4      4      4       K¾  	  \        P                  ! V4      # ©rï   )Úzipr-   rö   Úappendr   r   rú   Úaranger   rX   r=   ©r0   rV   rh  r&   ri  rE  Úquantrx  ry  ÚdrawÚk2s   &&&&&      r1   rT  Úhypergeom_gen._logsfÄ  sµ   € ØˆÜ&)¬2×+>Ò+>¸qÀQÓ+JÔ&KÑ"ˆE˜Ø˜•  c¥	Õ*¨d°S­j¸TÀC½ZÕ-HÔHà—
‘
œ5¤# d§k¡k°%¸dÓ&IÓ"JÐ!JÓKÖLô —Y’Y˜u q�y¨$°­(Ó3�Ø—
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œ9 T§\¡\°"¸4Ó%FÓGÖHñ 'Lô �zŠz˜#‹Ðr5   c                óà  € . p\        \        P                  ! WW44      !   Fµ  w  rgr‰VR ,           VR ,           ,          VR ,
          V	R ,
          ,          8”  d7   VP                  \	        \        V P                  WgW‰4      4      ) 4      4       Km  \        P                  ! ^ V^,           4      p
VP                  \        V P                  W§W‰4      4      4       K·  	  \        P                  ! V4      # r—  )r˜  r-   rö   r™  r   r   Úlogsfrš  r   rX   r=   r›  s   &&&&&      r1   rû   Úhypergeom_gen._logcdfÐ  s±   € ØˆÜ&)¬2×+>Ò+>¸qÀQÓ+JÔ&KÑ"ˆE˜Ø˜•  c¥	Õ*¨d°S­j¸TÀC½ZÕ-HÔHà—
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œ5¤# d§j¡j°¸TÓ&HÓ"IÐ!IÓJÖKô —Y’Y˜q %¨!¥)Ó,�Ø—
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œ9 T§\¡\°"¸4Ó%FÓGÖHñ 'Lô �zŠz˜#‹Ðr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rN   rH   rX   r^   rc   r…   r¸   rh   rT  rû   r�   r‘   r’   s   @r1   rf  rf  C  sO   ø‡ € ñGòPCô
Aò8òòò.ò.ò
ò"*ò
-ò
÷
ð 
r5   rf  Ú	hypergeomc                   óR   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# )Únhypergeom_genià  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                ó´   € \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      \        RR^ \        P                  3R4      .# )rh  Tr&   Úrr(   r,   r/   s   &r1   r2   Únhypergeom_gen._shape_infoD  rk  r5   c                ó
   € ^ V3# rE   r‡   )r0   rh  r&   r§  s   &&&&r1   rN   Únhypergeom_gen._get_supportI  rÉ   r5   c                óº   € V^ 8¬  W!8*  ,          V^ 8¬  ,          W1V,
          8*  ,          pV\        V4      \        V4      ,          \        V4      ,          ,          pV# rE   rF   )r0   rh  r&   r§  rù   s   &&&& r1   rH   Únhypergeom_gen._argcheckL  sK   € Ø�Q‘˜1™6Õ" a¨1¡fÕ-°¸µc±Õ:ˆØ”˜A“¤¨Q£Õ/´+¸a³.Õ@Õ@ˆØˆr5   Nc                ó8   a € \         V 3R  l4       pV! WW4VR7      # )c                 ó(  <€ SP                  WV4      w  rV\        P                  ! WV^,           4      pSP                  WpW4      p\	        W‡RRR7      p	V	! VP                  VR7      4      P                  \        4      p
Vf   V
P                  4       # V
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          ^,           W,
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          # rQ   rw  )rV   rh  r&   r§  s   &&&&r1   Ú<lambda>Ú(nhypergeom_gen._logpmf.<locals>.<lambda>d  s�   € Ü˜˜1�˜a“.�¤6¨!­#¨q£>Õ1Ü˜!�#˜a�% ¥ Q¥ q¥Ó)õ*Ü,2°1µ3°qµ5¸µ7¸AÓ,>õ?ä˜!˜A�#˜q�s 1�uÓ%õ&ä(.¨q°­s°A«ö7r5   ç        r$  )r6  r7  ©r0   rV   rh  r&   r§  s   &&&&&r1   rX   Únhypergeom_gen._logpmfa  s3   € Ü�ŠØ�!‰V˜˜Q™Õ !¨ ñ8ð ôð 	r5   c                ó8   € \        V P                  WW44      4      # rK   rÑ   rÁ  s   &&&&&r1   r^   Únhypergeom_gen._pmfj  s   € ô �4—<‘<  aÓ+Ó,Ð,r5   c                ó<  € R V,          R V,          R V,          r2pW2,          W,
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          ,          pRRrvWEWg3# )rÖ   Nr‡   )r0   rh  r&   r§  r|   r}   r~   r   s   &&&&    r1   r…   Únhypergeom_gen._statso  sv   € ð �Q•$˜˜1�˜b �dˆaˆØ�S�A•C˜•E�]ˆà�1•�g�a�i˜A�C �E A¥C¨¥E�?Õ+¨q°1¸!½¸A½µ;­Õ?ˆð �tˆBØ˜ˆÐr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rN   rH   rB   rX   r^   r…   r�   r‘   r’   s   @r1   r¥  r¥  à  s6   ø‡ € ñaòFCò
òô
Dò ò-÷
ð r5   r¥  Ú
nhypergeomc                   óL   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# )Ú
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                ó    € \        R RRR4      .# r™   rš   r/   s   &r1   r2   Úlogser_gen._shape_infoš  rœ   r5   Nc                ó&   € VP                  WR 7      # r?  )Ú	logseriesrŸ   s   &&&&r1   rB   Úlogser_gen._rvs�  s   € ð ×%Ñ% aÐ%Ó3Ð3r5   c                ó    € V^ 8„  V^8  ,          # rE   r‡   r¢   s   &&r1   rH   Úlogser_gen._argcheck¢  s   € Ø�A‘˜!˜a™%Õ Ð r5   c                ó„   € \         P                  ! W!4      ) R ,          V,          \        P                  ! V) 4      ,          # r   )r-   rK  r   r   rL  s   &&&r1   r^   Úlogser_gen._pmf¥  s,   € ä—’˜“ˆ Õ$ qÕ(¬7¯=ª=¸!¸Ó+<Õ<Ð<r5   c                óÂ   € R p\         P                  ! V^,           W24      ) \         P                  ! V^,           V4      ,          \        P                  ! V) 4      ,          # )g0Žä.ÿ++)r   rñ   rÃ   r-   r   )r0   rV   r'   Útinys   &&& r1   rh   Úlogser_gen._sf©  sF   € Øˆô
 —’  !¥ TÓ-Ð-´·²¸Q¸q½SÀ$Ó0GÕGÌ"Ï(Ê(ÐTUÐSUË,ÕVÐVr5   c                óP  € \         P                  ! V) 4      pWR ,
          ,          V,          pV) V,          VR ,
          ^,          ,          pWCV,          ,
          pV) V,          R V,           ,          R V,
          ^,          ,          pV^V,          V,          ,
          ^V^,          ,          ,           pV\        P                  ! VR4      ,          pV) V,          R V^,
          ^,          ,          ^V,          V^,
          ^,          ,          ,
          ^V,          V,          V^,
          ^,          ,          ,           ,          p	V	^V,          V,          ,
          ^V,          V,          V,          ,           ^V^,          ,          ,
          p
W¥^,          ,          R,
          pW5W‹3# )rÖ   ç      ø?r*  )r   r   r-   rK  )r0   r'   r§  r|   Úmu2pr}   Úmu3pÚmu3r~   Úmu4pÚmu4r   s   &&          r1   r…   Úlogser_gen._stats±  s4  € Ü�MŠM˜1˜"ÓˆØ�c•'�]˜QÕˆØˆr�A�v˜˜S� 1�Õ$ˆØ˜•U�lˆØˆr�A�v˜˜Q�Õ 3¨¥7¨Q¥,Õ.ˆØ�Q�r•T˜$•YÕ  2 q¥5¥Õ(ˆØ”2—8’8˜C Ó%Õ%ˆàˆr�A�vØ�1�Q•3˜•(�N˜Q˜q�S A¨¥E¨A¥:Õ-Õ-°°!µ°Aµ¸¸1½¸q½Õ0@Õ@õBˆà�Q�t•V˜B•YÕ  4¥¨¥¨2¥Õ-°°"°aµ%µÕ7ˆØ˜•6�\˜CÕˆØ˜ˆÐr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   r^   rh   r…   r�   r‘   r’   s   @r1   rÉ  rÉ  �  s.   ø‡ € ñò0>ô4ò
!ò=òW÷ð r5   rÉ  ÚlogserzA logarithmicc                   ó^   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# )Úpoisson_geniÄ  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                ó@   € \        R R^ \        P                  3R4      .# )r|   Fr(   r,   r/   s   &r1   r2   Úpoisson_gen._shape_infoÝ  s   € Ü˜4 ¨¬B¯F©F¨°]ÓCÐDÐDr5   c                ó   € V^ 8¬  # rE   r‡   )r0   r|   s   &&r1   rH   Úpoisson_gen._argchecká  s   € Ø�Q‰wˆr5   Nc                ó$   € VP                  W4      # rK   ©Úpoisson)r0   r|   r@   rA   s   &&&&r1   rB   Úpoisson_gen._rvsä  s   € Ø×#Ñ# BÓ-Ð-r5   c                ón   € \         P                  ! W4      \        V^,           4      ,
          V,
          pV# rQ   )r   rS   rR   )r0   rV   r|   ÚPks   &&& r1   rX   Úpoisson_gen._logpmfç  s'   € Ü�]Š]˜1Ó!¤E¨!¨a­%£LÕ0°2Õ5ˆØˆ	r5   c                ó6   € \        V P                  W4      4      # rK   rÑ   )r0   rV   r|   s   &&&r1   r^   Úpoisson_gen._pmfë  s   € ä�4—<‘< Ó&Ó'Ð'r5   c                óD   € \        V4      p\        P                  ! W24      # rK   )r   r   Úpdtr©r0   rU   r|   rV   s   &&& r1   rc   Úpoisson_gen._cdfï  s   € Ü�!‹HˆÜ�|Š|˜AÓ"Ð"r5   c                óD   € \        V4      p\        P                  ! W24      # rK   )r   r   Úpdtrcrð  s   &&& r1   rh   Úpoisson_gen._sfó  s   € Ü�!‹HˆÜ�}Š}˜QÓ#Ð#r5   c                óÚ   € \        \        P                  ! W4      4      p\        P                  ! V^,
          ^ 4      p\        P
                  ! WB4      p\        P                  ! WQ8¬  WC4      # rQ   )r   r   Úpdtrikr-   rq  rï  rD  )r0   rr   r|   rY  Úvals1rZ  s   &&&   r1   rs   Úpoisson_gen._ppf÷  sJ   € Ü”G—N’N 1Ó)Ó*ˆÜ—
’
˜4 !�8 QÓ'ˆÜ�|Š|˜EÓ&ˆÜ�xŠx˜™	 5Ó/Ð/r5   c                óä   € Tp\         P                  ! V4      pV^ 8„  p\        P                  ! WCR \         P                  R7      p\        P                  ! WCR \         P                  R7      pWWV3# )r   c                 ó&   € \        R V ,          4      # r   r+  ©rU   s   &r1   r¾  Ú$poisson_gen._stats.<locals>.<lambda>  s   € ¼¸SÀ½U¼r5   r$  c                 ó   € R V ,          # r   r‡   rû  s   &r1   r¾  rü    s   € ¸¸A¾r5   )r-   r=   r6  r7  r.   )r0   r|   r}   ÚtmpÚ
mu_nonzeror~   r   s   &&     r1   r…   Úpoisson_gen._statsý  sW   € ØˆÜ�jŠj˜‹nˆØ˜1‘Wˆ
Ü�_Š_˜ZÑ.CÔPR×PVÑPVÔWˆÜ�_Š_˜Z©oÌ"Ï&É&ÔQˆØ˜ˆÐr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rH   rB   rX   r^   rc   rh   rs   r…   r�   r‘   r’   s   @r1   rà  rà  Ä  s=   ø‡ € ñò0Eòô.òò(ò#ò$ò0÷ð r5   rà  rç  z	A Poisson)r•   rd  c                   ód   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
# )Ú
planck_geni	  aÙ  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                ó@   € \        R R^ \        P                  3R4      .# )Úlambda_Fr¿   r,   r/   s   &r1   r2   Úplanck_gen._shape_info$  s   € Ü˜9 e¨a´·±¨[¸.ÓIÐJÐJr5   c                ó   € V^ 8„  # rE   r‡   )r0   r  s   &&r1   rH   Úplanck_gen._argcheck'  s   € Ø˜‰{Ðr5   c                óL   € \        V) 4      ) \        V) V,          4      ,          # rK   )r   r   )r0   rV   r  s   &&&r1   r^   Úplanck_gen._pmf*  s    € Ü�w�h“Ð¤ W H¨Q¥J£Õ/Ð/r5   c                óN   € \        V4      p\        V) V^,           ,          4      ) # rQ   )r   r   ©r0   rU   r  rV   s   &&& r1   rc   Úplanck_gen._cdf-  s#   € Ü�!‹HˆÜ�w�h  !¥•nÓ%Ð%Ð%r5   c                ó6   € \        V P                  W4      4      # rK   )r   rT  )r0   rU   r  s   &&&r1   rh   Úplanck_gen._sf1  s   € Ü�4—;‘;˜qÓ*Ó+Ð+r5   c                ó:   € \        V4      pV) V^,           ,          # rQ   ©r   r  s   &&& r1   rT  Úplanck_gen._logsf4  s   € Ü�!‹HˆØˆx˜˜1��~Ðr5   c                óö   € \        RV,          \        V) 4      ,          ^,
          4      pV^,
          P                  ! V P                  V4      !  pV P	                  WB4      p\
        P                  ! WQ8¬  WC4      # )rÖ   ç      ð¿)r   r   ÚcliprN   rc   r-   rD  )r0   rr   r  rY  r÷  rZ  s   &&&   r1   rs   Úplanck_gen._ppf8  s^   € Ü�D˜•L¤5¨!¨£9Õ,¨QÕ.Ó/ˆØ�a•—’ × 1Ñ 1°'Ó :Ñ<ˆØ�y‰y˜Ó(ˆÜ�xŠx˜™	 5Ó/Ð/r5   Nc                óN   € \        V) 4      ) pVP                  WBR 7      R,
          # )r@  rÖ   )r   rA  )r0   r  r@   rA   r'   s   &&&& r1   rB   Úplanck_gen._rvs>  s)   € ä�G�8‹_ÐˆØ×%Ñ% aÐ%Ó3°cÕ9Ð9r5   c                óÚ   € ^\        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…   Úplanck_gen._statsC  sZ   € ØŒu�W‹~ÕˆÜ�7�(‹mœU G 8›_¨qÕ0Õ0ˆØŒt�G˜C•KÓ Õ ˆØˆq”�g“�ÕˆØ˜ˆÐr5   c                óp   € \        V) 4      ) pV\        V) 4      ,          V,          \        V4      ,
          # rK   )r   r   r   )r0   r  ÚCs   && r1   r¸   Úplanck_gen._entropyJ  s/   € Ü�G�8‹_ÐˆØ”s˜G˜8“}Õ$ QÕ&¬¨Q«Õ/Ð/r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rH   r^   rc   rh   rT  rs   rB   r…   r¸   r�   r‘   r’   s   @r1   r  r  	  sB   ø‡ € ñò4Kòò0ò&ò,òò0ô:ò
÷0ð 0r5   r  ÚplanckzA discrete exponential c                   óN   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# )Ú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                óz   € \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )r  Fri  Tr¿   r,   r/   s   &r1   r2   Úboltzmann_gen._shape_infoh  s:   € Ü˜9 e¨a´·±¨[¸.ÓIÜ˜3  q¬"¯&©& k°>ÓBðDð 	Dr5   c                ó@   € V^ 8„  V^ 8„  ,          \        V4      ,          # rE   rF   ©r0   r  ri  s   &&&r1   rH   Úboltzmann_gen._argcheckl  s   € Ø˜!‘  A¡Õ&¬°Q«Õ7Ð7r5   c                ó,   € V P                   V^,
          3# rQ   rL   r$  s   &&&r1   rN   Úboltzmann_gen._get_supporto  s   € Ø�v‰v�q˜1•uˆ}Ðr5   c                óš   € ^\        V) 4      ,
          ^\        V) V,          4      ,
          ,          pV\        V) V,          4      ,          # rQ   ©r   )r0   rV   r  ri  Úfacts   &&&& r1   r^   Úboltzmann_gen._pmfr  s<   € ð ”#�w�h“-• !¤C¨¨°­
£OÕ"3Õ4ˆØ”C˜˜ �
“OÕ#Ð#r5   c                ó˜   € \        V4      p^\        V) V^,           ,          4      ,
          ^\        V) V,          4      ,
          ,          # rQ   )r   r   )r0   rU   r  ri  rV   s   &&&& r1   rc   Úboltzmann_gen._cdfx  s9   € Ü�!‹HˆØ”#�w�h  !¥•nÓ%Õ%¨¬#¨w¨h°q­j«/Õ(9Õ:Ð:r5   c                óH  € V^\        V) V,          4      ,
          ,          p\        RV,          \        ^V,
          4      ,          ^,
          4      pV^,
          P                  R\        P
                  4      pV P                  WbV4      p\        P                  ! Wq8¬  We4      # )r*   rÀ  r  )r   r   r   r  r-   r.   rc   rD  )r0   rr   r  ri  ÚqnewrY  r÷  rZ  s   &&&&    r1   rs   Úboltzmann_gen._ppf|  sv   € Ø�!”C˜˜ �
“OÕ#Õ$ˆÜ�D˜•L¤3 q¨¥v£;Õ.¨qÕ0Ó1ˆØ�a•—‘˜c¤2§6¡6Ó*ˆØ�y‰y˜¨Ó+ˆÜ�xŠx˜™	 5Ó/Ð/r5   c                ó"  € \        V) 4      p\        V) V,          4      pVR V,
          ,          W$,          ^V,
          ,          ,
          pVR V,
          ^,          ,          W",          V,          ^V,
          ^,          ,          ,
          p^V,
          ^V,
          ,          pW7^,          ,          W",          V,          ,
          pV^V,           ,          V^,          ,          V^,          V,          ^V,           ,          ,
          p	W˜R,          ,          p	V^^V,          ,           W3,          ,           ,          V^,          ,          V^,          V,          ^^V,          ,           WD,          ,           ,          ,
          p
W¨,          V,          p
WVWš3# )rÖ   r×  r)  )r0   r  ri  ÚzÚzNr|   r}   ÚtrmÚtrm2r~   r   s   &&&        r1   r…   Úboltzmann_gen._statsƒ  s  € Ü��‹MˆÜ�'�˜!•‹_ˆØ��A•�Y�q•t˜Q˜r�T•{Õ"ˆØ��Q•˜•
�l˜Q�S �V Q r¥T¨A¥IÕ-Õ-ˆØ��t�a˜•c�lˆØ�q•&•˜1�3˜r�6Õ!ˆØ��!•�W�S˜!•V�^˜a �d 2�g q¨¥t�nÕ,ˆØ˜•+ÕˆØ��!�A•#•�a•c•	�]˜3 �6Õ! A q¥D¨2¥I¨q°°2µ­v°bµe­|Õ$<Õ<ˆØ�Y˜ÕˆØ˜ˆÐr5   r‡   N)r‹   rŒ   r�   rŽ   r�   r2   rH   rN   r^   rc   rs   r…   r�   r‘   r’   s   @r1   r   r   R  s3   ø‡ € ñò*Dò8òò$ò;ò0÷ð r5   r   Ú	boltzmannz!A truncated discrete exponential )r•   rM   rd  c                   ó^   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
# )Ú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                ó¶   € \        R R\        P                  ) \        P                  3R4      \        RR\        P                  ) \        P                  3R4      .# )ÚlowTÚhighr¿   r,   r/   s   &r1   r2   Úrandint_gen._shape_infoÕ  sH   € Ü˜5 $¬"¯&©&¨´"·&±&Ð(9¸>ÓJÜ˜6 4¬2¯6©6¨'´2·6±6Ð):¸NÓKðMð 	Mr5   c                óJ   € W!8„  \        V4      ,          \        V4      ,          # rK   rF   ©r0   r;  r<  s   &&&r1   rH   Úrandint_gen._argcheckÙ  s   € Ø‘
œk¨#Ó.Õ.´¸TÓ1BÕBÐBr5   c                ó   € W^,
          3# rQ   r‡   r?  s   &&&r1   rN   Úrandint_gen._get_supportÜ  s   € Ø˜•Fˆ{Ðr5   c                óØ   € \         P                  ! V4      \         P                  ! V\         P                  R 7      V,
          ,          p\         P                  ! W8¬  W8  ,          VR4      # )r8   rÀ  )r-   Ú	ones_liker=   Úint64rD  )r0   rV   r;  r<  r'   s   &&&& r1   r^   Úrandint_gen._pmfß  sD   € ä�LŠL˜‹OœrŸzšz¨$´b·h±hÔ?À#ÕEÕFˆÜ�xŠx˜™ a¡hÕ/°°BÓ7Ð7r5   c                óP   € \        V4      pWB,
          R ,           W2,
          ,          # r   r  )r0   rU   r;  r<  rV   s   &&&& r1   rc   Úrandint_gen._cdfä  s   € Ü�!‹HˆØ•˜"• ¥Õ,Ð,r5   c                óÔ   € \        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   Úrandint_gen._ppfè  sR   € Ü�A �Õ$ sÕ*Ó+¨aÕ/ˆØ˜•—‘ Ó*ˆØ�y‰y˜ TÓ*ˆÜ�xŠx˜™	 5Ó/Ð/r5   c                ó   € \         P                  ! V4      \         P                  ! V4      rCW4,           R ,
          ^,          pW4,
          pWf,          ^,
          R,          pRpRWf,          R ,           ,          Wf,          R ,
          ,          p	WWW‰3# )rÖ   g      (@rÀ  g333333ó¿)r-   r=   )
r0   r;  r<  Úm2Úm1r|   Údr}   r~   r   s
   &&&       r1   r…   Úrandint_gen._statsî  sk   € Ü—’˜DÓ!¤2§:¢:¨c£?ˆBØ�g˜�m˜qÕ ˆØ�GˆØ�s�Q�w˜$ÕˆØˆØ˜�˜s�Õ# q¥s¨S¥yÕ1ˆØ˜ˆÐr5   Nc                óž  € \         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@  )Úotypes)	r-   r=   r@   r   Úbroadcast_toÚ	vectorizer   r9   r>   )r0   r;  r<  r@   rA   Úrandints   &&&&& r1   rB   Úrandint_gen._rvs÷  sŽ   € ä�:Š:�c‹?×Ñ 1Ô$¬¯ª°DÓ)9×)>Ñ)>À!Ô)Cä °4ÔCÐCàÒô
 —/’/ #Ó,ˆCÜ—?’? 4Ó.ˆDÜ—,’,œw¤|°\ÓBÜ')§x¢x´£} oô7ˆá�sÓ!Ð!r5   c                ó$   € \        W!,
          4      # rK   )r   r?  s   &&&r1   r¸   Úrandint_gen._entropy  s   € Ü�4•:‹Ðr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rH   rN   r^   rc   rs   r…   rB   r¸   r�   r‘   r’   s   @r1   r9  r9  •  s?   ø‡ € ñ=ò~MòCòò8ò
-ò0òô"÷"ð r5   r9  rT  z#A discrete uniform (random integer)c                   óF   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# )Ú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                ó@   € \        R R^\        P                  3R4      .# ©rM   Fr¿   r,   r/   s   &r1   r2   Úzipf_gen._shape_info=  ó   € Ü˜3 ¨¬2¯6©6 {°NÓCÐDÐDr5   Nc                ó&   € VP                  WR 7      # r?  )Úzipf)r0   rM   r@   rA   s   &&&&r1   rB   Úzipf_gen._rvs@  s   € Ø× Ñ  Ð Ó.Ð.r5   c                ó   € V^8„  # rQ   r‡   ©r0   rM   s   &&r1   rH   Úzipf_gen._argcheckC  s   € Ø�1‰uˆr5   c                óœ   € VP                  \        P                  4      pR \        P                  ! V^4      ,          W) ,          ,          pV# r   )r×   r-   Úfloat64r   Úzeta)r0   rV   rM   rê  s   &&& r1   r^   Úzipf_gen._pmfF  s7   € Ø�H‰H”R—Z‘ZÓ ˆà”7—<’<  1Ó%Õ%¨¨2­Õ-ˆØˆ	r5   c                óh   € \         P                  ! W!^,           8„  W!3R \        P                  R7      # )r*   c                 ót   € \         P                  ! W,
          ^4      \         P                  ! V ^4      ,          # rQ   )r   rf  )rM   r&   s   &&r1   r¾  Ú zipf_gen._munp.<locals>.<lambda>O  s!   € œŸš a¥e¨QÓ/´'·,²,¸qÀ!Ó2DÖDr5   r$  r5  )r0   r&   rM   s   &&&r1   Ú_munpÚzipf_gen._munpL  s*   € Ü�ŠØ�A•‰I˜�vÙDÜ—v‘vôð 	r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   rH   r^   rk  r�   r‘   r’   s   @r1   rY  rY    s*   ø‡ € ñ)òVEô/òò÷ð r5   rY  r_  zA Zipfc                   óN   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# )Úzipfian_geniV  aÜ  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                óz   € \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )rM   FTr&   r(   r¿   r,   r/   s   &r1   r2   Úzipfian_gen._shape_info‰  s:   € Ü˜3 ¨¬2¯6©6 {°MÓBÜ˜3  q¬"¯&©& k°>ÓBðDð 	Dr5   c           	     ó°   € V^ 8¬  V\         P                  ! \         P                  ! V^R4      4      P                  \         P                  R7      8H  ,          # )r   r8   iÿÿÿ)r-   r=   r  r×   rE  ©r0   rM   r&   s   &&&r1   rH   Úzipfian_gen._argcheck�  sG   € ð �a‘Ø”b—j’j¤§¢¨¨A¨yÓ!9Ó:×AÑAÌÏÉÐAÓQÑQõSð 	Tr5   c                ó2   € ^\         P                  ! V4      3# rQ   )r-   r   rr  s   &&&r1   rN   Úzipfian_gen._get_supportž  s   € Ø”"—(’(˜1“+ˆ~Ðr5   c                óˆ   € \         P                  ! V4      p\         P                  ! V4      p\        P                  ! WW24      # rK   ©r-   r   r[   Ú_normalized_gen_harmonic©r0   rV   rM   r&   s   &&&&r1   r^   Úzipfian_gen._pmf¡  s/   € Ü�HŠH�Q‹KˆÜ�HŠH�Q‹KˆÜ×+Ò+¨A°!Ó7Ð7r5   c                óŠ   € \         P                  ! V4      p\         P                  ! V4      p\        P                  ! ^WV4      # rQ   rw  ry  s   &&&&r1   rc   Úzipfian_gen._cdf¦  s1   € Ü�HŠH�Q‹KˆÜ�HŠH�Q‹KˆÜ×+Ò+¨A¨q°QÓ7Ð7r5   c                ó˜   € \         P                  ! V4      p\         P                  ! V4      p\        P                  ! V^,           W3V4      # rQ   rw  ry  s   &&&&r1   rh   Úzipfian_gen._sf«  s5   € Ü�HŠH�Q‹KˆÜ�HŠH�Q‹KˆÜ×+Ò+¨A°­E°1¸Ó;Ð;r5   c                ód  € \         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
Wš,          pWc,          ^V,          V,          V^,          ,          ,
          ^V^,          ,          V^,          ,          ,           VR,          ,          pV^,          V,          ^V^,          ,          V,          V,          ,
          ^V,          V^,          ,          V,          ,           ^V^,          ,          ,
          V	^,          ,          pV^,          pW‹WÍ3# )r*   r×  )r-   r   r[   Ú_gen_harmonic)r0   rM   r&   ÚHnaÚHna1ÚHna2ÚHna3ÚHna4Úmu1Úmu2nÚmu2dÚmu2r~   r   s   &&&           r1   r…   Úzipfian_gen._stats°  s;  € Ü�HŠH�Q‹Kˆä×Ò Ó%ˆÜ× Ò   a¥CÓ(ˆÜ× Ò   a¥CÓ(ˆÜ× Ò   a¥CÓ(ˆÜ× Ò   a¥CÓ(ˆØ�hˆØ•˜4 �7Õ"ˆØ�A�vˆØ�kˆØ�h˜˜4� � S¨!¥VÕ+Õ+¨a°°aµ­i¸¸Q½Õ.>Õ>ÀÀcÅ
ÕJˆØ�1�f�T�k˜A˜c 1�f�H T�M¨$Õ.Õ.°°3µ°t¸Qµwµ¸tÕ1CÕCØ�$˜•'•	õØ! 1�Wõ%ˆà
ˆa�ˆØ˜ÐÐr5   r‡   N)r‹   rŒ   r�   rŽ   r�   r2   rH   rN   r^   rc   rh   r…   r�   r‘   r’   s   @r1   rn  rn  V  s5   ø‡ € ñ0òdDòTò"ò8ò
8ò
<÷
 ð  r5   rn  Úzipfianz	A Zipfianc                   óR   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	# )Údlaplace_geniÆ  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                ó@   € \        R R^ \        P                  3R4      .# r[  r,   r/   s   &r1   r2   Údlaplace_gen._shape_infoÝ  r]  r5   c                óh   € \        VR ,          4      \        V) \        V4      ,          4      ,          # ©rw   )r   r   Úabs)r0   rV   rM   s   &&&r1   r^   Údlaplace_gen._pmfà  s$   € ä�A�c•E‹{œS ! ¤c¨!£f¥Ó-Õ-Ð-r5   c                ó\   € \        V4      pR  pR p\        P                  ! V^ 8¬  W23WE4      # )c                 ód   € R \        V) V ,          4      \        V4      ^,           ,          ,
          # r   r)  ©rV   rM   s   &&r1   rò   Údlaplace_gen._cdf.<locals>.f1ç  s$   € Øœ˜a˜R !�V›¬¨A«°­
Õ3Õ3Ð3r5   c                 ó`   € \        W^,           ,          4      \        V4      ^,           ,          # rQ   r)  r–  s   &&r1   Úf2Údlaplace_gen._cdf.<locals>.f2ê  s    € Ü�q �E•{Ó#¤s¨1£v°¥zÕ2Ð2r5   )r   r6  r7  )r0   rU   rM   rV   rò   r™  s   &&&   r1   rc   Údlaplace_gen._cdfä  s0   € Ü�!‹Hˆò	4ò	3ô �Š˜q A™v¨ v¨rÓ6Ð6r5   c           
     óz  € ^\        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*   rÖ   )r   r   r-   rD  r   rc   )r0   rr   rM   ÚconstrY  r÷  s   &&&   r1   rs   Údlaplace_gen._ppfï  s�   € Ø”C˜“F•
ˆÜ”B—H’H˜Q ¨¬C°°«G­Õ!4Ñ4Ü  ¥›\¨AÕ-°Õ1Ü! 1 Q¥3¨%¥-Ó0Ð0°1Õ4ó6ó 7ˆð �q•ˆÜ�xŠx˜Ÿ	™	 %Ó+¨qÑ0°%Ó>Ð>r5   c                ó
  € \        V4      pR V,          VR,
          ^,          ,          pR V,          V^,          RV,          ,           R,           ,          VR,
          ^,          ,          pRVRWC^,          ,          R,
          3# )rw   rÖ   g      $@rÀ  r*  r)  )r0   rM   Úear‰  rÜ  s   &&   r1   r…   Údlaplace_gen._stats÷  se   € Ü�‹VˆØ��e�R˜•U˜Q•JÕˆØ��e�R˜•U˜3˜r�6•\ "•_Õ%¨¨B­°­
Õ2ˆØ�3˜˜C Q¥�J¨�OÐ+Ð+r5   c                óf   € V\        V4      ,          \        \        VR ,          4      4      ,
          # r‘  )r   r   r   rb  s   &&r1   r¸   Údlaplace_gen._entropyý  s"   € Ø”4˜“7�{œS¤ a¨¥e£Ó-Õ-Ð-r5   Nc                ó²   € \         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   ÚprobOfSuccessrU   Úys   &&&&   r1   rB   Údlaplace_gen._rvs   sL   € ô  Ÿš¤2§:¢:¨a£= .Ó1Ð1ˆØ×"Ñ" =Ð"Ó<ˆØ×"Ñ" =Ð"Ó<ˆØ�uˆr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   r^   rc   rs   r…   r¸   rB   r�   r‘   r’   s   @r1   r�  r�  Æ  s3   ø‡ € ñò,Eò.ò	7ò?ò,ò.÷ò r5   r�  ÚdlaplacezA discrete Laplacianc                   ól   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# )Úpoisson_binom_geni  uü  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 â‰¤ p_i â‰¤ 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                ó   € . # rK   r‡   r/   s   &r1   r2   Úpoisson_binom_gen._shape_infoH  s	   € ð ˆ	r5   c                ó€   € \         P                  ! V^ R7      p^ V8*  V^8*  ,          p\         P                  ! V^ R7      # )r   rŒ  )r-   Ústackr:   )r0   Úargsr'   Úcondss   &*  r1   rH   Úpoisson_binom_gen._argcheckM  s5   € Ü�HŠH�T Ô"ˆØ�a‘˜A ™FÕ#ˆÜ�vŠv�e !Ô$Ð$r5   r@   NrA   c               óF  € \         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*   rŒ  rž   éÿÿÿÿrQ   )	r-   r®  ÚshapeÚisscalarÚtupleÚbroadcast_shapesrº   rB   r‘  )r0   r@   rA   r¯  r'   s   &$$* r1   rB   Úpoisson_binom_gen._rvsR  s|   € ä�HŠH�T Ô#ˆð  š<�—’ÜŸ[š[¨×.Ò.��q‘	´E¸$³KÀ$Õ4Fð 	ä×"Ò" 1§7¡7¨DÓ1ˆÜ�~‰~˜a¸ˆ~ÓF×JÑJÐPRÐJÓSÐSr5   c                ó   € ^ \        V4      3# rE   )Úlen)r0   r¯  s   &*r1   rN   Úpoisson_binom_gen._get_support\  s   € Ø”#�d“)ˆ|Ðr5   c                óþ   € \         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�  r³  )r-   Ú
atleast_1dr×   rE  rö   r®  re  r   ©r0   rV   r¯  r'   s   &&* r1   r^   Úpoisson_binom_gen._pmf_  óW   € Ü�MŠM˜!Ó×#Ñ#¤B§H¡HÓ-ˆÜ×&Ò& qÐ0¨4Ó0ˆˆÜ�HŠH�T¤§¡°"Ô5ˆÜ! !Ó'Ð'r5   c                óþ   € \         P                  ! V4      P                  \         P                  4      p\         P                  ! V.VO5!  vr\         P
                  ! V\         P                  RR7      p\        W4      # r½  )r-   r¾  r×   rE  rö   r®  re  r   r¿  s   &&* r1   rc   Úpoisson_binom_gen._cdfe  rÁ  r5   c                ó¸   € \         P                  ! V^ R7      p\         P                  ! V^ R7      p\         P                  ! V^V,
          ,          ^ R7      pWERR3# )r   rŒ  N)r-   r®  r‘  )r0   r¯  Úkwdsr'   r"  r}   s   &*,   r1   r…   Úpoisson_binom_gen._statsk  sG   € Ü�HŠH�T Ô"ˆÜ�vŠv�a˜aÔ ˆÜ�fŠf�Q˜!˜A�#•Y QÔ'ˆØ˜4 Ð&Ð&r5   c                ó    € \        V .VO5/ VB # rK   )Úpoisson_binomial_frozen)r0   r¯  rÅ  s   &*,r1   Ú__call__Úpoisson_binom_gen.__call__q  s   € Ü& tÐ;¨dÒ;°dÑ;Ð;r5   c                ó>   <€ V ^8„  d   Qh/ S[ ;R&   S[ ;R&   S[ ;R&   # )r)  Ú_parse_args_rvsÚ_parse_args_statsÚ_parse_argsr   )Úformatr“   s   "€r1   Ú__annotate__Úpoisson_binom_gen.__annotate__  s2   ø‡ ‚ ñT  ÑñU ñV "Ñ!ñW ñX ÑòY r5   r‡   )r‹   rŒ   r�   rŽ   r�   r2   rH   rB   rN   r^   rc   r…   rÉ  Ú__annotate_func__r�   r‘   r’   s   @r1   rª  rª    sL   ø‡ € ñ'òZò
%ð
T˜tð T°$ô Tòò(ò(ò'ò<÷o ƒ r5   rª  Úpoisson_binomzA Poisson binomialr'   )r•   rd  Úshapesc                 óL   € \        \        P                  ! VR^ 4      4      VRV3# ©r*   rÖ   r³  ©r¶  r-   Úmoveaxis)r0   r'   Úlocr@   s   &&&&r1   rÌ  rÌ  }  s#   € Ü”—’˜Q  AÓ&Ó'¨¨c°4Ð7Ð7r5   c                 óL   € \        \        P                  ! VR^ 4      4      VRV3# rÖ  r×  )r0   r'   rÙ  r{   s   &&&&r1   rÍ  rÍ  €  s#   € Ü”—’˜Q  AÓ&Ó'¨¨c°7Ð:Ð:r5   c                 óJ   € \        \        P                  ! VR^ 4      4      VR3# rÖ  r×  )r0   r'   rÙ  s   &&&r1   rÎ  rÎ  ƒ  s!   € Ü”—’˜Q  AÓ&Ó'¨¨cÐ1Ð1r5   c                   ó0   a € ] tR tRt o R tRR ltRtV tR# )rÈ  i�  c                ó  € W n         W0n        VP                  ! R/ VP                  4       B V n        \
        P                  \        \        4      V P                  n        \        P                  \        \        4      V P                  n	        \        P                  \        \        4      V P                  n
        V P                  P                  ! V/ VB w  p pV P                  P                  ! V!  w  V n        V n        R # )Nr‡   )r¯  rÅ  Ú	__class__Ú_updated_ctor_paramÚdistrÌ  Ú__get__Ú_pb_objÚ_pb_clsrÍ  rÎ  rN   rM   r¥   )r0   rà  r¯  rÅ  rÔ  Ú_s   &&*,  r1   Ú__init__Ú poisson_binomial_frozen.__init__�  s²   € ØŒ	ØŒ	ð —N’NÑ@ T×%=Ñ%=Ó%?Ñ@ˆŒ	ô %4×$;Ñ$;¼GÄWÓ$Mˆ�	‰	Ô!Ü&7×&?Ñ&?ÄÌÓ&Qˆ�	‰	Ô#Ü +× 3Ñ 3´G¼WÓ Eˆ�	‰	Ôà—y‘y×,Ò,¨dÐ;°dÑ;‰ˆ��1ØŸ™×/Ò/°Ñ8‰ˆŒ�–r5   Nc                óº   € V P                   P                  ! V P                  / V P                  B w  rgpV P                   P                  ! WP                  WrW43/ VB # rK   )rà  rÎ  r¯  rÅ  Úexpect)	r0   ÚfuncÚlbÚubÚconditionalrÅ  rM   rÙ  Úscales	   &&&&&,   r1   rè  Úpoisson_binomial_frozen.expectž  sK   € ØŸ	™	×-Ò-¨t¯y©yÐF¸D¿I¹IÑF‰ˆ�ð �y‰y×Ò §i¡i°¸"ÑRÈTÑRÐRr5   )rM   r¯  r¥   rà  rÅ  )NNNF)r‹   rŒ   r�   rŽ   rå  rè  r�   r‘   r’   s   @r1   rÈ  rÈ  �  s   ø‡ € ò9÷Sò Sr5   rÈ  c                   óF   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# )Ú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                óz   € \        R R^ \        P                  3R4      \        RR^ \        P                  3R4      .# )r†  Fr‰  r¿   r,   r/   s   &r1   r2   Úskellam_gen._shape_infoÊ  s:   € Ü˜5 %¨!¬R¯V©V¨°nÓEÜ˜5 %¨!¬R¯V©V¨°nÓEðGð 	Gr5   Nc                óT   € TpVP                  W4      VP                  W%4      ,
          # rK   ræ  )r0   r†  r‰  r@   rA   r&   s   &&&&& r1   rB   Úskellam_gen._rvsÎ  s-   € ØˆØ×$Ñ$ SÓ,Ø×$Ñ$ SÓ,õ-ð 	.r5   c                ó˜  € \         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[   Ú	_ncx2_pdf©r0   rU   r†  r‰  Úpxs   &&&& r1   r^   Úskellam_gen._pmfÓ  sŒ   € Ü�[Š[˜h×'Ö'Ü—’˜!˜a™%ÜŸ-š-¨¨#­¨q°!°Aµ#­w¸¸#½Ó>¸qÕ@ÜŸ-š-¨¨#­¨q°!°Aµ#­w¸¸#½Ó>¸qÕ@óBˆB÷ (ð
 ˆ	÷ (Ö'ð
 ˆ	ús    BB8Â8C		c                ó„  € \        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)rô   r  Néþÿÿÿ)r   r-   r÷   rD  r   Úchndtrr[   Ú_ncx2_sfr÷  s   &&&& r1   rc   Úskellam_gen._cdfÛ  s†   € Ü�!‹HˆÜ�[Š[˜h×'Ö'Ü—’˜!˜a™%Ü!Ÿ.š.¨¨3­°°1µ°a¸µeÓ<ÜŸ,š, q¨¥u¨a°°1µ­g°q¸µuÓ=ó?ˆB÷ (ð ˆ	÷	 (Ö'ð ˆ	ús   «A9B.Â.B?	c                ón   € W,
          pW,           pV\        V^,          4      ,          p^V,          pW4WV3# )é   r+  )r0   r†  r‰  r"  r}   r~   r   s   &&&    r1   r…   Úskellam_gen._statsã  s6   € Ø�yˆØ�iˆØ”D˜# �“NÕ"ˆØ��WˆØ˜"Ð Ð r5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   r^   rc   r…   r�   r‘   r’   s   @r1   rð  rð  ¥  s*   ø‡ € ñ#òHGô.ò
ò÷!ð !r5   rð  Úskellamz	A Skellamc                   ó^   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# )Úyulesimon_geniî  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                ó@   € \        R R^ \        P                  3R4      .# )ÚalphaFr¿   r,   r/   s   &r1   r2   Úyulesimon_gen._shape_info  s   € Ü˜7 E¨A¬r¯v©v¨;¸ÓGÐHÐHr5   Nc           	     ó¦   € VP                  V4      pVP                  V4      p\        V) \        \        V) V,          4      ) 4      ,          4      pV# rK   )Ústandard_exponentialr   r   r   )r0   r  r@   rA   ÚE1ÚE2Úanss   &&&&   r1   rB   Úyulesimon_gen._rvs  sK   € Ø×.Ñ.¨tÓ4ˆØ×.Ñ.¨tÓ4ˆÜ�B�3œ¤ R C¨%¥KÓ 0Ð0Ó1Õ1Ó2ˆØˆ
r5   c                óJ   € V\         P                  ! W^,           4      ,          # rQ   ©r   rÃ   ©r0   rU   r  s   &&&r1   r^   Úyulesimon_gen._pmf  s   € Ø”w—|’| A¨q¥yÓ1Õ1Ð1r5   c                ó   € V^ 8„  # rE   r‡   )r0   r  s   &&r1   rH   Úyulesimon_gen._argcheck  s   € Ø˜‘	Ðr5   c                ó\   € \        V4      \        P                  ! W^,           4      ,           # rQ   ©r   r   r	   r  s   &&&r1   rX   Úyulesimon_gen._logpmf  s   € Ü�5‹zœGŸNšN¨1°a­iÓ8Õ8Ð8r5   c                óX   € ^V\         P                  ! W^,           4      ,          ,
          # rQ   r  r  s   &&&r1   rc   Úyulesimon_gen._cdf"  s   € Ø�1”w—|’| A¨q¥yÓ1Õ1Õ1Ð1r5   c                óJ   € V\         P                  ! W^,           4      ,          # rQ   r  r  s   &&&r1   rh   Úyulesimon_gen._sf%  s   € Ø”7—<’< ¨1¥9Ó-Õ-Ð-r5   c                ó\   € \        V4      \        P                  ! W^,           4      ,           # rQ   r  r  s   &&&r1   rT  Úyulesimon_gen._logsf(  s   € Ü�1‹vœŸš q°!­)Ó4Õ4Ð4r5   c                óÎ  € \         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.   Únanr   )r0   r  r|   r‰  r~   r   s   &&    r1   r…   Úyulesimon_gen._stats+  sQ  € Ü�XŠX�e˜q‘j¤"§&¡&¨%¸1µ9Õ*=Ó>ˆÜ�hŠh�u˜q‘yØ˜a•x E¨C¥K°E¸AµIÀµ>Õ#AÕBÜ—v‘vóˆô �hŠh�u ‘z¤2§6¡6¨3Ó/ˆÜ�XŠX�e˜a‘iÜ˜5 1�9“o¨°­°Q­Õ6¸%È1Å9Õ:MÕNÜ—f‘fóˆô �XŠX�e˜q‘j¤"§&¡&¨"Ó-ˆÜ�XŠX�e˜a‘iØ˜a•i B¨°­¥M°B¸µJÕ$>ÀÕ$CØ$)°Q­YÕ$7¸5À1½9Õ$Eõ$Gõ Hä—f‘fóˆô �XŠX�e˜q‘j¤"§&¡&¨"Ó-ˆØ˜ˆÐr5   r‡   rˆ   )r‹   rŒ   r�   rŽ   r�   r2   rB   r^   rH   rX   rc   rh   rT  r…   r�   r‘   r’   s   @r1   r  r  î  s>   ø‡ € ñ òBIôò2òò9ò2ò.ò5÷ð r5   r  Ú	yulesimon)r•   rM   c                   óX   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# )Ú_nchypergeom_geni@  z�A noncentral hypergeometric discrete random variable.

For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.

Nc           	     óî   € \        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  Úoddsr(   r¿   r,   r/   s   &r1   r2   Ú_nchypergeom_gen._shape_infoJ  sf   € Ü˜3  q¬"¯&©& k°=ÓAÜ˜3  q¬"¯&©& k°=ÓAÜ˜3  q¬"¯&©& k°=ÓAÜ˜6 5¨1¬b¯f©f¨+°~ÓFðHð 	Hr5   c                ó†   € YTr%pW5,
          p\         P                  ! ^ W&,
          4      p\         P                  ! W%4      pWx3# rE   rp  )	r0   rh  r&   ri  r$  rM  rL  Úx_minÚx_maxs	   &&&&&    r1   rN   Ú_nchypergeom_gen._get_supportP  s:   € Ø˜ˆqˆØ�VˆÜ—
’
˜1˜a�fÓ%ˆÜ—
’
˜1Ó!ˆØˆ|Ðr5   c                ó„  € \         P                  ! V4      \         P                  ! V4      r!\         P                  ! V4      \         P                  ! V4      rC\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8¬  ,          p\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8¬  ,          p\         P                  ! V4      ( VP                  \        4      V8H  ,          V^ 8¬  ,          pV^ 8„  pW18*  p	W!8*  p
WV,          V,          V,          V	,          V
,          # rE   )r-   r=   Úisnanr×   r>   )r0   rh  r&   ri  r$  Úcond1Úcond2Úcond3Úcond4Úcond5Úcond6s   &&&&&      r1   rH   Ú_nchypergeom_gen._argcheckW  sâ   € Ü�zŠz˜!‹}œbŸjšj¨›mˆ1Ü—*’*˜Q“-¤§¢¨DÓ!1ˆ4Ü—(’(˜1“+� !§(¡(¬3£-°1Ñ"4Õ5¸¸a¹Õ@ˆÜ—(’(˜1“+� !§(¡(¬3£-°1Ñ"4Õ5¸¸a¹Õ@ˆÜ—(’(˜1“+� !§(¡(¬3£-°1Ñ"4Õ5¸¸a¹Õ@ˆØ�q‘ˆØ‘ˆØ‘ˆØ�}˜uÕ$ uÕ,¨uÕ4°uÕ<Ð<r5   c           	     ó8   a € \         V 3R  l4       pV! WW4WVR7      # )c                 ó˜  <€ \         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+  Úfullr  Úprodr!   ÚgetattrÚrvs_nameÚreshape)rh  r&   ri  r$  r@   rA   ÚlengthÚurnÚrv_genr·  r0   s   &&&&&&    €r1   r¸  Ú$_nchypergeom_gen._rvs.<locals>._rvs1d  s†   ø€ ä�xŠx˜‹{œRŸXšX a›[Õ(¬2¯8ª8°A«;×6Ô6Ü—w’w˜t¤R§V¡VÓ,Ð,Ü—W’W˜T“]ˆFÜ#Ó%ˆCÜ˜S $§-¡-Ó0ˆFÙ˜˜q¨Ó=ˆCØ—+‘+˜dÓ#ˆCØˆJr5   rž   rº  )r0   rh  r&   ri  r$  r@   rA   r¸  s   f&&&&&& r1   rB   Ú_nchypergeom_gen._rvsb  s&   ø€ ä	#ô	ó 
$ð	ñ �Q˜1¨ÔIÐIr5   c                óÐ   a € \         P                  ! WW4V4      w  rr4pVP                  ^ 8X  d   \         P                  ! V4      # \         P                  V 3R l4       pV! WW4V4      # )r   c                 óF  <€ \         P                  ! V 4      \         P                  ! V4      ,          \         P                  ! V4      ,          \         P                  ! V4      ,          '       d   \         P                  # SP                  W2WR 4      pVP	                  V 4      # ©gê-�™—q=)r-   r+  r  rà  Úprobability)rU   rh  r&   ri  r$  r;  r0   s   &&&&& €r1   Ú_pmf1Ú$_nchypergeom_gen._pmf.<locals>._pmf1w  sb   ø€ ä�xŠx˜‹{œRŸXšX a›[Õ(¬2¯8ª8°A«;Õ6¼¿ºÀ!»×DÔDÜ—v‘v�Ø—)‘)˜A !¨5Ó1ˆCØ—?‘? 1Ó%Ð%r5   )r-   rö   r@   Ú
empty_likerS  )r0   rU   rh  r&   ri  r$  rC  s   f&&&&& r1   r^   Ú_nchypergeom_gen._pmfq  s_   ø€ ä×.Ò.¨q°Q¸4Ó@Ñˆˆa�DØ�6‰6�QŒ;Ü—=’= Ó#Ð#ä	�‰ô	&ó 
ð	&ñ �Q˜1 Ó&Ð&r5   c                óz   a € \         P                  V 3R  l4       pRV9   g   RV9   d
   V! WW44      MRw  rxRRr©WxWš3# )c                 ó.  <€ \         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  rà  r{   )rh  r&   ri  r$  r;  r0   s   &&&& €r1   Ú	_moments1Ú*_nchypergeom_gen._stats.<locals>._moments1‚  s[   ø€ ä�xŠx˜‹{œRŸXšX a›[Õ(¬2¯8ª8°A«;×6Ô6Ü—v‘vœrŸv™v�~Ð%Ø—)‘)˜A !¨5Ó1ˆCØ—;‘;“=Ð r5   r‰  ÚvNrˆ   )r-   rS  )r0   rh  r&   ri  r$  r{   rI  r‰  rK  rv   rV   s   f&&&&&     r1   r…   Ú_nchypergeom_gen._stats€  sL   ø€ ä	�‰ô	!ó 
ð	!ð .1°G¬^¸sÀg¼~‘	˜! Ô(Ø!ñ 	ˆà�Tˆ1Ø�QˆzÐr5   r‡   rˆ   r‰   )r‹   rŒ   r�   rŽ   r�   r8  rà  r2   rN   rH   rB   r^   r…   r�   r‘   r’   s   @r1   r"  r"  @  s;   ø‡ € ñð €HØ€DòHòò	=ôJò'÷ò r5   r"  c                   ó"   € ] tR tRtRtRt]tRtR# )Ú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

Ú
rvs_fisherr‡   N)	r‹   rŒ   r�   rŽ   r�   r8  r   rà  r�   r‡   r5   r1   rN  rN  �  s   † ñGðR €HØ%„Dr5   rN  Únchypergeom_fisherz$A Fisher's noncentral hypergeometricc                   ó"   € ] tR tRtRtRt]tRtR# )Únchypergeom_wallenius_geniâ  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

Úrvs_walleniusr‡   N)	r‹   rŒ   r�   rŽ   r�   r8  r    rà  r�   r‡   r5   r1   rR  rR  â  s   † ñGðR €HØ'„Dr5   rR  Únchypergeom_walleniusz&A Wallenius' noncentral hypergeometric)r   N)r   rŠ   rE   )lÚ	functoolsr   Útypesr   Úscipyr   Úscipy.specialr   r   r	   r
   rR   Úscipy.special._ufuncsÚ_ufuncsr[   Úscipy.special._spfun_statsr   r   Úscipy._lib._utilr   Úscipy._external.array_api_extraÚ	_externalÚarray_api_extrar6  Úscipy.interpolater   Únumpyr   r   r   r   r   r   r   r   r   r   r-   Ú_distn_infrastructurer   r   r   r   r   r   Ú
_biasedurnr   r    r!   r#   r”   r—   rº   r¼   rÛ   rÝ   r  r  r9  r;  rc  rf  r£  r¥  rÇ  rÉ  rÞ  rà  rç  r  r  r   r7  r9  rT  rY  r_  rn  r‹  r�  r.   r¨  rª  rÓ  rÌ  rÍ  rÎ  râ  rã  rá  rÈ  rð  r  r  r   r"  rN  rP  rR  rT  ÚlistÚglobalsÚcopyÚitemsÚpairsÚ_distn_namesÚ_distn_gen_namesÚ__all__r‡   r5   r1   Ú<module>rl     s‰  ðõ
 Ý å ß CÓ Cß #Ð #ß MÝ )ß -Ð -Ý &ç M× M× Mã ÷8÷ 8÷,ñ ,ô
Z�ô Zñz 	�wÔ€ô>#�Iô >#ñB ˜A KÔ0€	ôQ�Kô Qñh ˜{Ô+€	ôr
�ô r
ñj 
˜Ô	"€ôZ�[ô Zñz  Ô.€
ôM7ˆ{ô M7ñ` �!˜&¨=Ô9€ôW�Kô Wñt ˜{Ô+€	ôZ�[ô Zñz  Ô.€
ô=�ô =ñ@ 
�a˜h°Ô	A€ô?�+ô ?ñD ˜9¨{Ô
;€ôC0�ô C0ñL 
�a˜hÐ1JÔ	K€ô<�Kô <ñ~ ˜{¨aØ#FôH€	ôt�+ô tñn ˜9ð 0)ô *€ô
?ˆ{ô ?ñD �!˜&¨8Ô4€ôj �+ô j ñZ ˜ 	°KÔ
@€ôM�;ô Mñ` ˜2Ÿ6™6˜'Ø'Ð2HôJ€ôX<˜ô X<ñv " ÐAUØ),ô.€ô8ô;ô2ð
 !Ð"3Ð €ˆØ /× 7Ñ 7¸ÀÓ I€Ô Ø"3×";Ñ";¸GÀWÓ"M€Ô Ø'×/Ñ/°¸ÓA€Ô ôSÐ0ô Sô0C!�+ô C!ñL ˜Ÿ™˜ i¸+Ô
F€ôL�Kô Lñ^ ˜{¨aÔ0€	ôL�{ô Lô^K&Ð-ô K&ñ\ ,Ø	Ø3ô5Ð ô
K(Ð 0ô K(ñ\ 2Ø	 Ø5ô7Ð ñ 	‰W‹Y�^‰^Ó×#Ñ#Ó%Ó&€Ù!7¸¸{Ó!KÑ €Ðà
Ð)Õ
)‚r5   