Expectation of Weibull Distribution

Theorem
Let $X$ be a continuous random variable with the Weibull distribution with $\alpha, \beta \in \R_{> 0}$.

The expectation of $X$ is given by:


 * $\ds \expect {X} = \beta^1 \map \Gamma {1 + \frac 1 \alpha}$

where $\Gamma$ is the Gamma function.

Proof
From Raw Moment of Weibull Distribution, we have:

The $n$th raw moment $\expect {X^n}$ of $X$ is given by:


 * $\ds \expect {X^n} = \beta^n \map \Gamma {1 + \frac n \alpha}$

Therefore, for $n = 1$ we have:


 * $\ds \expect {X} = \beta^1 \map \Gamma {1 + \frac 1 \alpha}$