# Fisher’s Noncentral Hypergeometric Distribution¶

A random variable has Fisher’s 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

$p(x; M, n, N, \omega) = \frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},$

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

$P_k = \sum_{y=x_l}^{x_u} \binom{n}{y} \binom{M - n}{N-y} \omega^y y^k,$

and the binomial coefficients are

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

Other functions of this distribution are

\begin{eqnarray*} \mu & = & \frac{P_0}{P_1},\\ \mu_{2} & = & \frac{P_2}{P_0} - \left(\frac{P_1}{P_0}\right)^2,\\ \end{eqnarray*}

## References¶

Implementation: scipy.stats.nchypergeom_fisher