Expectation of Gamma Distribution/Proof 2

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


 * $\displaystyle M_X \left({t}\right) = \left({1 - \frac t \beta}\right)^{-\alpha}$

for $t < \beta$.

From Moment in terms of Moment Generating Function:


 * $\mathbb E \left[{X}\right] = M'_X \left({0}\right)$

We have:

Setting $t = 0$: