Second Derivative of PGF of Negative Binomial Distribution

Theorem

First Form

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

Then the second derivative of the PGF of $X$ with respect to $s$ is:

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

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 second derivative of the PGF of $X$ with respect to $s$ is:

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