Expectation of Binomial Distribution/Proof 3

Theorem
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

Then the expectation of $X$ is given by:
 * $E \left({X}\right) = n p$

Proof
From the Probability Generating Function of Binomial Distribution, we have:
 * $\displaystyle \Pi_X \left({s}\right) = \left({q + ps}\right)^n$

where $q = 1 - p$.

From Expectation of Discrete Random Variable from P.G.F., we have:
 * $\displaystyle E \left({X}\right) = \Pi'_X \left({1}\right)$

We have:

Plugging in $s = 1$:
 * $\displaystyle\Pi'_X \left({1}\right) = n p \left({q + p}\right)$

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