Probability Generating Function of Negative Binomial Distribution/First Form

Theorem

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:

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

where $q = 1 - p$.

Proof

From the definition of p.g.f:

$\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map {p_X} k s^k$

From the definition of the negative binomial distribution (first form):

$\map {p_X} k = \dbinom {n + k - 1} {n - 1} p^k q^n$

where $q = 1 - p$.

So:

\(\ds \map {\Pi_X} s\)

\(=\)

\(\ds \sum_{k \mathop \ge 0} \binom {n + k - 1} {n - 1} p^k q^n s^k\)

\(\ds \)

\(=\)

\(\ds q^n \sum_{k \mathop \ge 0} \binom {n + k - 1} {n - 1} \paren {p s}^k\)

\(\ds \)

\(=\)

\(\ds \paren {\frac q {1 - p s} }^n\)

For the third equality, the equation in the second line is rewritten in terms of binomial series.

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Hence the result.

$\blacksquare$