Expectation of Power of Gamma Distribution

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

Then:
 * $\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$

where:
 * $\expect {X^n}$ denotes the expectation of $X^n$
 * $\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$.

Proof
From Moment in terms of Moment Generating Function:


 * $\expect {X^n} = \map { {M_X}^{\paren n} } 0$

where ${M_X}^{\paren n}$ denotes the $n$th derivative of $M_X$.

Then: