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


 * $\map { {M_X}'''} t = \dfrac {\beta^\alpha \alpha \paren {\alpha + 1} \paren {\alpha + 2} } {\paren {\beta - t}^{\alpha + 3} }$

Proof
We have: