Moment Generating Function of Beta Distribution

Theorem
Let $X \sim \BetaDist \alpha \beta$ denote the Beta distribution fior some $\alpha, \beta > 0$.

Then the moment generating function $M_X$ of $X$ is given by:


 * $\ds \map {M_X} t = 1 + \sum_{k \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {t^k} {k!}$

Proof
From the definition of the Beta distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$

From the definition of a moment generating function:


 * $\ds \map {M_X} t = \expect {e^{t X} } = \int_0^1 e^{t x} \map {f_X} x \rd x$

So: