Negative Binomial Distribution

The negative binomial random variable with parameters \(n\) and \(p\in\left(0,1\right)\) can be defined as the number of extra independent trials (beyond \(n\) ) required to accumulate a total of \(n\) successes where the probability of a success on each trial is \(p.\) Equivalently, this random variable is the number of failures encoutered while accumulating \(n\) successes during independent trials of an experiment that succeeds with probability \(p.\) Thus,

\[ \begin{eqnarray*} p\left(k;n,p\right) & = & \left(\begin{array}{c} k+n-1\\ n-1\end{array}\right)p^{n}\left(1-p\right)^{k}\quad k\geq0\\ F\left(x;n,p\right) & = & \sum_{i=0}^{\left\lfloor x\right\rfloor }\left(\begin{array}{c} i+n-1\\ i\end{array}\right)p^{n}\left(1-p\right)^{i}\quad x\geq0\\ & = & I_{p}\left(n,\left\lfloor x\right\rfloor +1\right)\quad x\geq0\\ \mu & = & n\frac{1-p}{p}\\ \mu_{2} & = & n\frac{1-p}{p^{2}}\\ \gamma_{1} & = & \frac{2-p}{\sqrt{n\left(1-p\right)}}\\ \gamma_{2} & = & \frac{p^{2}+6\left(1-p\right)}{n\left(1-p\right)}.\end{eqnarray*}\]

Recall that \(I_{p}\left(a,b\right)\) is the incomplete beta integral.

Implementation: scipy.stats.nbinom