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

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

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Proof

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

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

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

Thus we have:

\(\ds \frac \d {\d s} \map {\Pi_X} s\)

\(=\)

\(\ds \frac \d {\d s} \paren {\frac {p s} {1 - q s} }^n\)

\(\ds \)

\(=\)

\(\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \frac \d {\d s} \paren {\frac {p s} {1 - q s} }\)

Power Rule for Derivatives and Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {\paren {1 - q s} \frac \d {\d s} \paren {p s} - p s \frac \d {\d s} \paren {1 - q s} } {\paren {1 - q s}^2} }\)

Quotient Rule for Derivatives and Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {\paren {1 - q s} p - p s \paren {-q} } {\paren {1 - q s}^2} }\)

Power Rule for Derivatives where $n = 1$

\(\ds \)

\(=\)

\(\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac {p - p q s + p q s} {\paren {1 - q s}^2} }\)

\(\ds \)

\(=\)

\(\ds n \paren {\frac {p s} {1 - q s} }^{n - 1} \paren {\frac p {\paren {1 - q s}^2} }\)

\(\ds \)

\(=\)

\(\ds n p \paren {\frac {\paren {p s}^{n - 1} } {\paren {1 - q s}^{n + 1} } }\)

Hence the result.

$\blacksquare$