Derivatives 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 derivatives of the PGF of $X$ with respect to $s$ are:
- $\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = ...$
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.
When this page/section has been completed, {{Stub}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Proof
The Probability Generating Function of Negative Binomial Distribution (Second Form) is:
- $\Pi_X \left({s}\right) = \left({\dfrac {p s} {1 - q s}}\right)^n$
We have that for a given negative binomial distribution , $n, p$ and $q$ are constant.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
