Expectation of Gamma Distribution/Proof 2

Proof
By Moment Generating Function of Gamma 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:


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

From Moment Generating Function of Gamma Distribution: First Moment:


 * $\map { {M_X}'} t = \dfrac {\beta^\alpha \alpha} {\paren {\beta - t}^{\alpha + 1} }$

Hence setting $t = 0$: