.. _discrete-nchypergeom-wallenius:

Wallenius' Noncentral Hypergeometric Distribution

A random variable has Wallenius' Noncentral Hypergeometric distribution with parameters

:math:M \in {\mathbb N}, :math:n \in [0, M], :math:N \in [0, M], :math:\omega > 0,

if its probability mass function is given by

.. 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], 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

.. math::

\binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.

References

Implementation: scipy.stats.nchypergeom_wallenius