Second Derivative of PGF of Negative Binomial Distribution/Second Form

Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.

Then the second derivative of the PGF of $X$ w.r.t. $s$ is:


 * $\dfrac {\mathrm d^2} {\mathrm d s^2} \Pi_X \left({s}\right) = \left({\dfrac {ps} {1 - qs} }\right)^{n+1} \left({\dfrac {n \left({n-1}\right) + 2 n q s} {p s^3 \left({1 - qs}\right)} }\right)$

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


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

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

From First Derivative of PGF of Negative Binomial Distribution/Second Form:

Thus we have: