First Derivative of PGF of Negative Binomial Distribution/Second Form

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Theorem

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

Then the first derivative of the PGF of $X$ with respect to $s$ is:

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

Proof

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

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


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


Thus we have:

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

Hence the result.

$\blacksquare$