First Derivative of PGF of Negative Binomial Distribution/First Form

From ProofWiki
Jump to: navigation, search

Theorem

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

Then the first derivative of the PGF of $X$ w.r.t. $s$ is:

$\dfrac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right) = \dfrac {n p} q \left({\dfrac q {1 - ps} }\right)^{n+1}$

Proof

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

$\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$


We have that for a given negative binomial distribution, $n, p$ and $q$ are constant.


Thus we have:

\(\displaystyle \frac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right)\) \(=\) \(\displaystyle \frac {\mathrm d} {\mathrm d s} \left({\dfrac q {1 - ps} }\right)^n\)
\(\displaystyle \) \(=\) \(\displaystyle n \left({\frac q {1 - ps} }\right)^{n-1} \frac {\mathrm d} {\mathrm d s} \left({\frac q {1 - ps} }\right)\) Power Rule for Derivatives and Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle n \left({\frac q {1 - ps} }\right)^{n-1} \left({-p}\right) \left({\frac {-q} {\left({1 - ps}\right)^2} }\right)\) Power Rule for Derivatives and Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle \frac {n p} q \left({\frac q {1 - ps} }\right)^{n+1}\) simplifying

Hence the result.

$\blacksquare$