ExponentialRandomVariable 2001-10-26 04:17:15 2004-04-09 12:43:44 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} $X$ is a \emph{exponential random variable} with parameter $\lambda>0$ if its probability density function is given for $x>0$ by \begin{align*} f_X(x) = \lambda e^{-\lambda x}. \end{align*} To denote this, one usually writes $X\sim Exp(\lambda)$. For an exponential random variable $X$: \begin{enumerate} \item $X$ is commonly used to model lifetimes and duration between Poisson events. \item The expected value of $X$ is given by $E[X] = \frac{1}{\lambda}$ \item The variance of $X$ is given by $Var[X] = \frac{1}{\lambda^2}$ \item The moments of $X$ are given by $M_X(t) = \frac{\lambda}{\lambda - t}$ \item It is interesting to note that $X$ is a gamma random variable with an $\alpha$ parameter of 1. \end{enumerate}