Variance 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:


 * $\operatorname{var} \left({X}\right) = \dfrac {\alpha \beta} {\left({\alpha + \beta}\right)^2 \left({\alpha + \beta + 1}\right)}$

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 Variance as Expectation of Square minus Square of Expectation:


 * $\displaystyle \operatorname{var} \left({X}\right) = \int_0^1 x^2 f_X \left({x}\right) \rd x - \left({\mathbb E \left[{X}\right]}\right)^2$

So: