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:


 * $\expect X = \dfrac {n p} q$

where $q = 1 - p$.

Proof
From Probability Generating Function of Negative Binomial Distribution (First Form):
 * $\map {\Pi_X} s = \paren {\dfrac q {1 - p s} }^n$

From Expectation of Discrete Random Variable from PGF:
 * $\expect X = \map {\Pi'_X} 1$

We have:

Plugging in $s = 1$: