Expectation of Exponential Distribution/Proof 2

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


 * $\map {M_X} t = \dfrac 1 {1 - \beta t}$

By Moment in terms of Moment Generating Function:


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

We have:

Setting $t = 0$ gives: