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

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

Equations

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

Links

• The Continuous Uniform Distribution
• The Discrete Uniform Distribution