Bayesian Statistics: An Introduction, 4th Edition

A.13 Negative binomial distribution

X has a negative binomial distribution of index n and parameter π, denoted

if it has a discrete distribution with density

Because

we sometimes use the notation

The mean and variance are

Because

we see that p(X+1)> p(X) if and only if

and hence that a mode occurs at

the square brackets denoting ‘integer part of’, and this mode is unique unless is an integer.

It can be shown that the distribution function can be found in terms of that of the binomial distribution, or equivalently in terms of the incomplete beta function; for details see Balakrishnan et ...