Expectation of Poisson Distribution/Proof 3

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


 * $\map {M_X} t = e^{\lambda \paren {e^t - 1} }$

By Moment in terms of Moment Generating Function:


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

We have:

Setting $t = 0$ gives: