Probability Generating Function of Negative Binomial Distribution

Theorem

Type 1

Let $X$ be a discrete random variable with the type 1 negative binomial distribution with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:

$\ds \map {\Pi_X} s = \paren {\frac {p s} {1 - q s} }^n$

where $q = 1 - p$.

Type 2

Let $X$ be a discrete random variable with the type $2$ negative binomial distribution with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:

$\map {\Pi_X} s = \paren {\dfrac q {1 - p s} }^n$

where $q = 1 - p$.