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

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

Equations

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_X(x)\delta x$	$=1$
$f_X(x)$	$=\{\begin{array}{cc}\frac{1}{b-a}& a<x<b\\ 0& \text{else}\end{array}$
$F_X(x)$	$=\{\begin{array}{cc}0& x<a\\ \frac{x-a}{b-a}& a\le x<b\\ 1& x\ge b\end{array}$
$h_X(x)$	$=???$
$M_X(x)$	$=\frac{e^{xb}-e^{xa}}{x(b-a)}$

Links

• The Continuous Uniform Distribution
• The Discrete Uniform Distribution