> [!NOTE] Theorem > Let $a<b.$ Let $X$ be a [[Continuous Uniform Distribution|uniformly distributed real-valued random variable]] on $(a,b).$ Then $X$ is [[Square-Integrable Continuous Real-Valued Random Variable|square-integrable]] and its [[Variance of a Square-Integrable Continuous Real-Valued Random Variable|variance]] is given by $\text{Var}(X)=\frac{(b-a)^{2}}{4}.$ **Proof**: By [[Expectation of Real-Valued Function of Continuous Real-Valued Random Variable]], $\mathbb{E}[X^{2}] = \frac{1}{b-a}\int_{a}^{b} x^{2} \, dx = \frac{b^{3}-a^{3}}{3(b-a)} = \frac{a^{2}+ab+b^{2}}{3}$ Thus by [[Alternative Formula for Variance of a Square-Integrable Continuous Real-Valued Random Variable]] and [[Expectation of Continuous Uniform Distribution]], $\text{Var}(X)=\mathbb{E}[X^{2}]-(\mathbb{E}[X] )^{2}= \frac{a^{2}+ab+b^{2}}{3} - \frac{a^{2}+2ab+b^{2}}{4} = \frac{(b-a)^{2}}{4} .$