Expectation of Gamma Distribution

Theorem
Let $X \sim \Gamma \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.

Then:


 * $\displaystyle \mathbb E \left[{X}\right] = \frac \alpha \beta$

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


 * $\displaystyle f_X\left({x}\right) = \frac{ \beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\Gamma \left({\alpha}\right)}$

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


 * $\displaystyle \mathbb E \left[{X}\right] = \int_0^\infty x f_X \left({x}\right) \rd x$

So: