Probability Generating Function of Negative Binomial Distribution/Second Form

Theorem

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:

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

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:

 $\displaystyle \Pi_X \left({s}\right)$ $=$ $\displaystyle \sum_{k \mathop \ge n} \binom {k-1} {n-1} p^n q^{k-n} s^k$ $\displaystyle$ $=$ $\displaystyle \frac {p^n}{q^n} \sum_{k \mathop \ge n} \binom {k-1} {n-1} \left({q s}\right)^k$ $\displaystyle$ $=$ $\displaystyle \left({\frac {ps} {1 - qs} }\right)^n$

Hence the result.

$\blacksquare$