Probability Generating Function of Negative Binomial Distribution/Second Form

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Theorem

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:

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


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.

$\blacksquare$


Sources