# Expectation of Negative Binomial Distribution/First Form

## Theorem

Then the expectation of $X$ is given by:

$E \left({X}\right) = \dfrac {n p} q$

## Proof

$\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$

where $q = 1 - p$.

From Expectation of Discrete Random Variable from PGF, we have:

$E \left({X}\right) = \Pi'_X \left({1}\right)$

We have:

 $\ds \Pi'_X \left({s}\right)$ $=$ $\ds \frac {\mathrm d} {\mathrm ds} \left({\dfrac q {1 - ps} }\right)^n$ $\ds$ $=$ $\ds \dfrac {n p} q \left({\dfrac q {1 - ps} }\right)^{n+1}$ First Derivative of PGF of Negative Binomial Distribution/First Form

Plugging in $s = 1$:

 $\ds \Pi'_X \left({1}\right)$ $=$ $\ds \frac {n p} q \left({\frac q {1 - p} }\right)^{n+1}$ $\ds$ $=$ $\ds \frac {n p} q \left({\frac q q }\right)^{n+1}$ as $p = 1 - q$ $\ds \implies \ \$ $\ds E \left({X}\right)$ $=$ $\ds \frac {n p} q$ simplifying

$\blacksquare$