UniformContinousRandomVariable 2001-10-26 04:05:12 2006-10-25 00:19:56 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A random variable $X$ is said to be a \emph{\PMlinkescapetext{uniform} (\PMlinkescapetext{continuous}) random variable} with parameters \textbf{$a$ and $b$} if its probability density function is given by \begin{align*} f_X(x) = \frac{1}{b-a},\quad\quad x \in [a,b], \end{align*} and is denoted $X\sim U(a,b)$.\\ Notes: \begin{enumerate} \item They are also called \emph{rectangular distributions}, considers that all points in the interval $[a,b]$ have the same mass. \item $E[X] = \frac{a+b}{2}$ \item $Var[X] = \frac{(b-a)^2}{12}$ \item $M_X(t) = \frac{e^{bt} - e^{at}}{(b-a)t}$ \end{enumerate}