.. _discrete-nchypergeom-fisher:

Fisher's Noncentral Hypergeometric Distribution

A random variable has Fisher's 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; 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], where :math:x_l = \max(0, N - (M - n)), :math:x_u = \min(N, n),

.. math::

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

.. math::

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

Other functions of this distribution are

.. math:: :nowrap:

\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