Moment Generating Function of Gamma Distribution/Examples/Second 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 second moment generating function of $X$ is given by:


 * $\ds \map { {M_X}''} t = \frac {\beta^\alpha \alpha \paren {\alpha + 1} } {\paren {\beta - t}^{\alpha + 2} }$

Proof
We have: