# Derivatives of PGF of Negative Binomial Distribution/Second Form

Jump to navigation
Jump to search

## 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:

- $\dfrac {\d^k} {\d s^k} \map {\Pi_X} s = ...$

This article is incomplete.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Stub}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |