Expectation of Beta Distribution/Proof 1

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: