## The Negative Binomial Distribution

The negative binomial models the number of trials\[n\]

up to and including a given number of successes \[x\]

, where the probability of success is a fixed \[p\]

. The first \[x-1\]

successes may be any of the first \[n-1\]

trials, but the \[n\]

th trial must be the \[x\]

th success.We may model the first

\[x-1\]

successes from \[n-1\]

by the binomial \[B(n-1,p)\]

distribution:\[\begin{equation} \begin{aligned}P(X=x-1) &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{(n-1)-(x-1)} \\ &={}^{n-1}C_{x-1}p^{x-1}(1-p)^{n-x}{}\end{aligned} \end{equation}\]

The

\[n\]

th trial is the \[x\]

th success with probability \[p\]

, so the probability of the needing \[n\]

trials to obtain \[x\]

successes is \[{}^{n-1}C_{x-1}p^x(1-p)^{n-x}\]

.It is important to realise that the variable being modelled here is

\[n\]

. \[x\]

is the fixed number of successes.It is often convenient to reparametrize in terms of number of successes and failures, so let

\[k\]

be number of failures, then \[x+k=n\]

and the expression becomes \[{}^{x+k-1}C_{x-1}p^x(1-p)^{k}\]

.