Variance 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 Variance as Expectation of Square minus Square of Expectation:


 * $\ds \var X = \int_0^1 x^2 \map {f_X} X \rd x - \paren {\expect X}^2$

So: