> [!NOTE] Theorem > Let $a<b.$ Let $X$ be a [[Continuous Uniform Distribution|uniformly distributed real-valued random variable]] on $(a,b).$ Then $X$ is [[Integrable Continuous Real-Valued Random Variable|integrable]] and its [[Expectation of Integrable Continuous Real-Valued Random Variable|expectation]] is given by $\mathbb{E}[X]=\frac{a+b}{2}.$ **Proof**: By definition, we have $\mathbb{E}[X] = \frac{1}{b-a} \int_{a}^{b} x \, dx = \frac{b^{2}-a^{2}}{2(b-a)} = \frac{a+b}{2} .$ # Applications By [[Variance of Continuous Uniform Distribution]], if $X\sim\mathcal{U}(a,b)$ then $Var(X)=\mathbb{E}[X^{2}]-(\mathbb{E}[X])^{2} = \frac{(b-a)^{2}}{4}.$