Expectation 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:


 * $\mathbb E \left[{X}\right] = \dfrac \alpha {\alpha + \beta}$

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 the expected value of a continuous random variable:


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

So: