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Wallenius' Noncentral Hypergeometric Distribution

docs/tutorial:stats:discrete_nchypergeom_wallenius

A random variable has Wallenius' 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; N, n, M) = (x n) (N - x M - n) \int_{0}^{1} (1 - t^{ω / D})^{x} (1 - t^{1/ D})^{N - x} d t

for $x \in [x_{l}, x_{u}]$ , where $x_{l} = max (0, N - (M - n))$ , $x_{u} = min (N, n)$ ,

D = ω (n - x) + ((M - n) - (N - x)),

and the binomial coefficients are

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

References

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

Implementation: scipy.stats.nchypergeom_wallenius