Moment Generating Function of Gamma Distribution/Examples/First Moment

Examples of Use of Moment Generating Function of Gamma Distribution
Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.

Let $t < \beta$.

The first moment generating function of $X$ is given by:


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

Proof
We have: