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

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


Proof

We have:

\(\ds \map { {M_X}''} t\) \(=\) \(\ds \frac {\d^2} {\d t^2} \paren {1 - \frac t \beta}^{-\alpha}\)
\(\ds \) \(=\) \(\ds \beta^\alpha \frac {\d^2} {\d t^2} \paren {\frac 1 {\paren {\beta - t}^\alpha} }\)
\(\ds \) \(=\) \(\ds \beta^\alpha \map {\frac \d {\d t} } {\map {\frac \d {\map \d {\beta - t} } } {\frac 1 {\paren {\beta - t}^\alpha} } \cdot \map {\frac \d {\d t} } {\beta - t} }\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \paren {-1}^2 \beta^\alpha \alpha \map {\frac \d {\d t} } {\frac 1 {\paren {\beta - t}^{\alpha + 1} } }\) Derivative of Power
\(\ds \) \(=\) \(\ds \frac {\beta^\alpha \alpha \paren {\alpha + 1} } {\paren {\beta - t}^{\alpha + 2} }\) Chain Rule for Derivatives, Derivative of Power

$\blacksquare$