Derivatives of PGF of Negative Binomial Distribution/Second Form

Theorem

Then the derivatives of the PGF of $X$ with respect to $s$ are:

$\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = ...$

Proof

$\Pi_X \left({s}\right) = \left({\dfrac {p s} {1 - q s}}\right)^n$

We have that for a given negative binomial distribution , $n, p$ and $q$ are constant.