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


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

So: