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 encountered 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