Derivatives of Moment Generating Function of Gamma Distribution

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Theorem

Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.

Let $t < \beta$.

Let $M_X$ denote the moment generating function of $X$.


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 $\alpha$.


Proof

The proof proceeds by induction on $n$.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

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


Basis for the Induction

$\map P 0$ is the case:

\(\ds {M_X}^{\paren 0}\) \(=\) \(\ds M_X\) Definition of Higher Derivative
\(\ds \) \(=\) \(\ds \paren {1 - \dfrac t \beta}^{-\alpha}\) Moment Generating Function of Gamma Distribution
\(\ds \) \(=\) \(\ds \paren {\dfrac {\beta - t} \beta}^{-\alpha}\)
\(\ds \) \(=\) \(\ds \dfrac {\beta^\alpha} {\paren {\beta - t}^\alpha}\)
\(\ds \) \(=\) \(\ds \dfrac {\alpha^{\overline 0} \beta^\alpha} {\paren {\beta - t}^{\alpha + 0} }\) Number to Power of Zero Rising is One


Thus $\map P 0$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is the induction hypothesis:

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


from which it is to be shown that:

${M_X}^{\paren {k + 1} } = \dfrac {\alpha^{\overline {k + 1} } \beta^\alpha} {\paren {\beta - t}^{\alpha + k + 1} }$


Induction Step

This is the induction step:

\(\ds {M_X}^{\paren {k + 1} }\) \(=\) \(\ds \dfrac \d {\d t} {M_X}^{\paren k}\) Definition of Higher Derivative
\(\ds \) \(=\) \(\ds \dfrac \d {\d t} \dfrac {\alpha^{\overline k} \beta^\alpha} {\paren {\beta - t}^{\alpha + k} }\) Induction Hypothesis
\(\ds \) \(=\) \(\ds \alpha^{\overline k} \beta^\alpha \dfrac \d {\d t} \paren {\beta - t}^{-\alpha - k}\)
\(\ds \) \(=\) \(\ds \alpha^{\overline k} \beta^\alpha \paren {-1} \paren {-\alpha - k} \paren {\beta - t}^{-\alpha - k - 1}\) Power Rule for Derivatives, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \alpha^{\overline k} \beta^\alpha \paren {\alpha + k} \dfrac 1 {\paren {\beta - t}^{\alpha + k + 1} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {\alpha^{\overline {k + 1} } \beta^\alpha} {\paren {\beta - t}^{\alpha + k + 1} }\) Definition of Rising Factorial

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \Z_{\ge 0}: {M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$

$\blacksquare$

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