Expectation of Beta Distribution/Proof 2

Theorem

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

Then:

$\expect X = \dfrac \alpha {\alpha + \beta}$

Proof

\(\ds \expect X\)

\(=\)

\(\ds \prod_{r \mathop = 0}^0 \frac {\alpha + r} {\alpha + \beta + r}\)

Raw Moment of Beta Distribution

\(\ds \)

\(=\)

\(\ds \frac {\alpha + 0} {\alpha + \beta + 0}\)

\(\ds \)

\(=\)

\(\ds \frac \alpha {\alpha + \beta}\)

$\blacksquare$