Moment Generating Function of Beta Distribution

Theorem
Let $X \sim \operatorname{Beta} \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.

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


 * $\displaystyle M_X \left({t}\right) = 1 + \sum_{k \mathop = 1}^\infty \left({ \prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} }\right) \frac{t^k} {k!}$

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


 * $\displaystyle f_X \left({x}\right) = \frac { x^{\alpha - 1} \left({1 - x}\right)^{\beta - 1} } {\Beta \left({\alpha, \beta}\right)}$

From the definition of a moment generating function:


 * $\displaystyle M_X \left({t}\right) = \mathbb E \left[{ e^{t X} }\right] = \int_0^1 e^{tx} f_X \left({x}\right) \rd x$

So: