First Derivative of PGF of Negative Binomial Distribution/First Form

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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:

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

Proof

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

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


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


Thus we have:

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

Hence the result.

$\blacksquare$