Variance of Beta Distribution

Theorem
Let $X \sim \map \Beta {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Beta$ is the Beta distribution.

Then:


 * $\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$

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:


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

So: