# University of Alberta Guide/STAT/222/Uniform Random Variables

Uniform Random Variables are used where each outcome of an event is equally probable. For ${\displaystyle p_{U}(u)=P(U\leq u)}$ where ${\displaystyle U\,}$ is some ${\displaystyle [a,b]\,}$-Uniform Random Variable, ${\displaystyle p_{U}(u)\,}$ will always be zero unless ${\displaystyle (a\leq u\leq b)}$. When ${\displaystyle (a\leq u\leq b)}$, then ${\displaystyle p_{U}(u)={\frac {1}{b-a}}}$.

## Equations

 ${\displaystyle \int _{-\infty }^{\infty }f_{X}(x)\delta x}$ ${\displaystyle =1\,}$ ${\displaystyle f_{X}(x)\,}$ ${\displaystyle ={\begin{cases}{\frac {1}{b-a}}&a ${\displaystyle F_{X}(x)\,}$ ${\displaystyle ={\begin{cases}0&x ${\displaystyle h_{X}(x)\,}$ ${\displaystyle =???\,}$ ${\displaystyle M_{X}(x)\,}$ ${\displaystyle ={\frac {e^{xb}-e^{xa}}{x\left(b-a\right)}}}$