Moment Generating Function of Gamma Distribution/Examples/Fourth Moment

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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 fourth moment generating function of $X$ is given by:

$\map { {M_X}^{\paren 4} } t = \dfrac {\beta^\alpha \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} } {\paren {\beta - t}^{\alpha + 4} }$


Proof

We have:

\(\ds \map { {M_X}^{\paren 4} } t\) \(=\) \(\ds \frac \d {\d t} \map { {M_X}'''} t\) Definition of Moment Generating Function
\(\ds \) \(=\) \(\ds \frac \d {\d t} \frac {\beta^\alpha \alpha \paren {\alpha + 1} \paren {\alpha + 2} } {\paren {\beta - t}^{\alpha + 3} }\) Moment Generating Function of Gamma Distribution: Third Moment
\(\ds \) \(=\) \(\ds \paren {-1} \beta^\alpha \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} {\frac {-1} {\paren {\beta - t}^{\alpha + 4} } }\) Chain Rule for Derivatives, Derivative of Power
\(\ds \) \(=\) \(\ds \dfrac {\beta^\alpha \alpha \paren {\alpha + 1} \paren {\alpha + 2} \paren {\alpha + 3} } {\paren {\beta - t}^{\alpha + 4} }\)

$\blacksquare$