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:
 * $$\Pi_X \left({s}\right) = \left({\frac {q} {1 - ps}}\right)^n$$

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:
 * $$\Pi_X \left({s}\right) = \left({\frac {ps} {1 - qs}}\right)^n$$

where $$q = 1 - p$$.

Proof
From the definition of p.g.f:


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

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

where $$q = 1 - p$$.

So:

$$ $$ $$

Hence the result.

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

where $$q = 1 - p$$.

So:

$$ $$ $$

Hence the result.