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 ...
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