Expectation of Beta Distribution/Proof 1

Proof
From the definition of the beta distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$

From the definition of the expected value of a continuous random variable:


 * $\displaystyle \expect X = \int_0^1 x \, \map {f_X} x \rd x$

So: