Raw Moment of Beta Distribution

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

Then:


 * $\displaystyle \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$

for positive integer $n$.

Proof
By Moment Generating Function of Beta Distribution, the moment generating function, $M_X$, of $X$ is given by:


 * $\displaystyle \map {M_X} t = \expect {e^{t X} } = 1 + \sum_{n \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac{t^n} {n!}$

We also have:

Comparing coefficients yields the result.