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 $X$ with respect to $s$ is:
- $\dfrac \d {\d s} \map {\Pi_X} s = \dfrac {n p} q \paren {\dfrac q {1 - p s} }^{n + 1}$
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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:
\(\ds \frac \d {\d s} \map {\Pi_X} s\) | \(=\) | \(\ds \map {\frac \d {\d s} } {\paren {\dfrac q {1 - p s} }^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {\frac q {1 - p s} }^{n - 1} \map {\frac \d {\d s} } {\frac q {1 - p s} }\) | Power Rule for Derivatives and Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds n \paren {\frac q {1 - p s} }^{n - 1} \paren {-p} \paren {\frac {-q} {\paren {1 - p s}^2} }\) | Power Rule for Derivatives and Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {n p} q \paren {\frac q {1 - p s} }^{n + 1}\) | simplifying |
Hence the result.
$\blacksquare$