Expectation of Negative Binomial Distribution/First Form

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

Then the expectation of $X$ is given by:


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

Proof
From Probability Generating Function of Negative Binomial Distribution (First Form), we have:
 * $\Pi_X \left({s}\right) = \left({\dfrac q {1 - ps}}\right)^n$

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:

Plugging in $s = 1$: