Variance of Gamma Distribution/Proof 1

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


 * $\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$

From Variance as Expectation of Square minus Square of Expectation:


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

So: