Probability Generating Function of Negative Binomial Distribution

First Form
Let $X$ be a discrete random variable with the negative binomial distribution (first form) with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:
 * $\displaystyle \Pi_X \left({s}\right) = \left({\frac {q} {1 - ps}}\right)^n$

where $q = 1 - p$.

Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.

Then the p.g.f. of $X$ is:
 * $\displaystyle \Pi_X \left({s}\right) = \left({\frac {ps} {1 - qs}}\right)^n$

where $q = 1 - p$.

Proof
From the definition of p.g.f:


 * $\displaystyle \Pi_X \left({s}\right) = \sum_{k \ge 0} p_X \left({k}\right) s^k$

Proof of First Form
From the definition of the negative binomial distribution (first form):
 * $\displaystyle p_X \left({k}\right) = \binom {n + k - 1} {n - 1} p^k q^n$

where $q = 1 - p$.

So:

Hence the result.

Proof of Second Form
From the definition of the negative binomial distribution (second form):
 * $\displaystyle p_X \left({k}\right) = \binom {k-1} {n-1} p^n q^{k-n}$

where $q = 1 - p$.

So:

Hence the result.