## Mean and Variance of Negative Binomial Distribution

\[n\]

to achieve a desired number \[x\]

of successes.We can treat the negative binomial as a sequence of geometric sequences each with parameter

\[p\]

, each sequence ending in a success. The expected number of trial until a success is achieved in \[\frac{1}{p}\]

and since we require \[x\]

successes we would expect to have to carry out \[\frac{x}{p}\]

trials.Similarly the variance of a geometric distribution is

\[\frac{1-p}{p^2}\]

so the variance of a sequence of \[x\]

such geometric distributions is \[\frac{x(1-p)}{p^2}\]

.