Derivatives of PGF of Negative Binomial Distribution
Jump to navigation
Jump to search
Theorem
First Form
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:
- $\dfrac {\d^k} {\d s^k} \map {\Pi_X} s = \dfrac {n^{\overline k} p^k} {q^k} \paren {\dfrac q {1 - p s} }^{n + k}$
where:
- $n^{\overline k}$ is the rising factorial: $n^{\overline k} = n \paren {n + 1} \paren {n + 2} \cdots \paren {n + k - 1}$
- $q = 1 - p$
Second Form
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.
Then the derivatives of the PGF of $X$ with respect to $s$ are:
- $\dfrac {\d^k} {\d s^k} \map {\Pi_X} s = ...$