Uniform Continuous Probability Distribution

Definition

$X\sim \text{Uniform}(u,l)$ takes values in interval $[l,u],l<u\in \mathbb{R}$ s.t probability of any subinterval $[a,b]$ is proportional to its length: $P(a<X<b)=\frac{b-a}{u-l},\forall l\le a\le b\le u$

PDF Requirement

$$\{\begin{array}{ll}\frac{1}{u-l}& l\le x\le u\\ 0& \text{otherwise}\end{array}$$

CDF Requirement

$$\{\begin{array}{ll}0& x<l\\ \frac{x-l}{u-l}& l\le x\le u\\ 1& x>u\end{array}$$

Expectation

$E(X)=\frac{u+l}{2}$

Variance

$V(X)=\frac{(u-l{)}^2}{12}$

MGF

$$\frac{e^{tu}-e^{tl}}{t(u-l)}$$