Expectation 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 expectation of $X$ is given by:


 * $\expect X = \dfrac n p$

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

where $q = 1 - p$.

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

We have:

Plugging in $s = 1$: