Derivatives of PGF 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 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

The Probability Generating Function of Negative Binomial Distribution (Second Form) is:

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


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