Raw Moment 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:


 * $\displaystyle \mathbb E \left[{X^n}\right] = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$

for positive integer $n$.

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


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

We also have:

Comparing coefficients yields the result.