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Fisher's Noncentral Hypergeometric Distribution

docs/tutorial:stats:discrete_nchypergeom_fisher

A random variable has Fisher's Noncentral Hypergeometric distribution with parameters

$M \in N$ , $n \in [0, M]$ , $N \in [0, M]$ , $ω > 0$ ,

if its probability mass function is given by

p (x; M, n, N, ω) = \frac{( x n ) ( N - x M - n ) ω ^{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} = y = x_{l} \sum x_{u} (y n) (N - y M - n) ω^{y} y^{k},

and the binomial coefficients are

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

Other functions of this distribution are

References

Agner Fog, "Biased Urn Theory", https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf
"Fisher's noncentral hypergeometric distribution", Wikipedia, https://en.wikipedia.org/wiki/Fisher's_noncentral_hypergeometric_distribution

Implementation: scipy.stats.nchypergeom_fisher