> [!NOTE] Definition (Continuous Uniform Distribution) > Let $X$ be a [[Continuous random variables|continuous real-valued random variable]]. Let $a <b.$ Then $X$ is said to be uniformly distributed on the [[Real intervals|interval]] $(a,b)$, denoted $X\sim\mathcal{U}(a,b),$ if its [[Probability Density Function|probability density function]] is given by $f_{X}= \frac{1}{b-a} \mathbb{1}_{[a,b]}(x) = \begin{cases} \frac{1}{b-a} & a\leq x\leq b \\ 0 &\text{otherwise} \end{cases}$ # Properties By [[Expectation of Continuous Uniform Distribution]], if $X\sim\mathcal{U}(a,b)$ then $\mathbb{E}[X]=\frac{a+b}{2}.$ By [[Variance of Continuous Uniform Distribution]], if $X\sim\mathcal{U}(a,b)$ then $Var(X)=\frac{(b-a)^{2}}{4}.$