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

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

Proof

We have:

\(\ds \map { {M_X}'} t\)

\(=\)

\(\ds \map {\frac \d {\d t} } {1 - \frac t \beta}^{-\alpha}\)

Moment Generating Function of Gamma Distribution

\(\ds \)

\(=\)

\(\ds \beta^\alpha \map {\frac \d {\d t} } {\frac 1 {\paren {\beta - t}^\alpha} }\)

\(\ds \)

\(=\)

\(\ds \beta^\alpha \, \map {\frac \d {\map \d {\beta - t} } } {\frac 1 {\paren {\beta - t}^\alpha} } \cdot \map {\frac \d {\d t} } {\beta - t}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \paren {-1}^2 \frac {\beta^\alpha \alpha} {\paren {\beta - t}^{\alpha + 1} }\)

Derivative of Power

\(\ds \)

\(=\)

\(\ds \frac {\beta^\alpha \alpha} {\paren {\beta - t}^{\alpha + 1} }\)

$\blacksquare$