# Wallenius’ Noncentral Hypergeometric Distribution¶

A random variable has Wallenius’ Noncentral Hypergeometric distribution with parameters

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

if its probability mass function is given by

$p(x; N, n, M) = \binom{n}{x} \binom{M - n}{N-x}\int_0^1 \left(1-t^{\omega/D}\right)^x\left(1-t^{1/D}\right)^{N-x} dt$

for $$x \in [x_l, x_u]$$, where $$x_l = \max(0, N - (M - n))$$, $$x_u = \min(N, n)$$,

$D = \omega(n - x) + ((M - n)-(N-x)),$

and the binomial coefficients are

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

## References¶

Implementation: scipy.stats.nchypergeom_wallenius