Expectation of Binomial Distribution/Proof 3

Proof
From the Probability Generating Function of Binomial Distribution, we have:
 * $\map {\Pi_X} s = \paren {q + p s}^n$

where $q = 1 - p$.

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

We have:

Plugging in $s = 1$:
 * $\map {\Pi'_X} 1 = n p \paren {q + p}$

Hence the result, as $q + p = 1$.