Moment Generating Function of Gamma Distribution/Examples

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 $t < \beta$.

First Moment

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

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

Second Moment

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

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

Third Moment

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

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

Fourth Moment

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

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

Derivatives of Moment Generating Function of Gamma Distribution‎

The $n$th derivative of $M_X$ is given by:

${M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$

where $\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$.