First Derivative of PGF of Negative Binomial Distribution/First Form

Theorem

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

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

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.

$\blacksquare$