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Negative Hypergeometric Distribution

Consider a box containing \(M\) balls: \(n\) red and \(M-n\) blue. We randomly sample balls from the box, one at a time and without replacement, until we have picked \(r\) blue balls. nhypergeom is the distribution of the number of red balls \(k\) we have picked.

\begin{eqnarray*} p(k;M,n,r) & = & \frac{\left(\begin{array}{c} k+r-1\\ k\end{array}\right)\left(\begin{array}{c} M-r-k\\ n-k\end{array}\right)}{\left(\begin{array}{c} M\\ n\end{array}\right)}\quad 0 \leq k \leq M-n,\\ F(x;M,n,r) & = & \sum_{k=0}^{\left\lfloor x\right\rfloor }p\left(k;M,n,r\right),\\ \mu & = & \frac{rn}{M-n+1},\\ \mu_{2} & = & \frac{rn(M+1)}{(M-n+1)(M-n+2)}\left(1-\frac{r}{M-n+1}\right) \end{eqnarray*}

for \(k \in 0, 1, 2, ..., n\), where the binomial coefficients are defined as,

\begin{eqnarray*} \binom{n}{k} \equiv \frac{n!}{k! (n - k)!} \end{eqnarray*}

The cumulative distribution, survivor function, hazard function, cumulative hazard function, inverse distribution function, moment generating function, and characteristic function on the support of \(k\) are mathematically intractable.

Implementation: scipy.stats.nhypergeom

Wallenius’ Noncentral Hypergeometric Distribution Zipf (Zeta) Distribution