Expectation of Gamma Distribution/Proof 2

Proof
By Moment Generating Function of Gaussian Distribution, the moment generating function of $X$ is given by:


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

for $t < \beta$.

From Moment in terms of Moment Generating Function:


 * $\expect X = \map {M'_X} 0$

We have:

Setting $t = 0$: