Expectation of Binomial Distribution/Proof 4

Proof
From Moment Generating Function of Binomial Distribution, the moment generating function of $X$, $M_X$, is given by:


 * $\displaystyle \map {M_X} t = \paren {1 - p + pe^t}^n$

By Moment in terms of Moment Generating Function:


 * $\displaystyle \expect X = \map {M_X'} 0$

We have:

Setting $t = 0$ gives: