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 $X$ $s$ is:


 * $\dfrac {\d^2} {\d s^2} \map {\Pi_X} s = \paren {\dfrac {p s} {1 - q s} }^{n + 2} \paren {\dfrac {n \paren {n - 1} + 2 n q s} {\paren {p s^2}^2} }$

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


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

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: