CauchyRandomVariable 2001-10-26 04:44:25 2007-06-24 00:55:11 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak} \newcommand{\ol}{\overline} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\La}{\Leftarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\nor}{\vartriangleleft} \newcommand{\Gal}{\text{Gal}} \newcommand{\GL}{\text{GL}} \newcommand{\Z}{\mb{Z}} \newcommand{\R}{\mb{R}} \newcommand{\Q}{\mb{Q}} \newcommand{\C}{\mb{C}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} $X$ is a \textbf{Cauchy random variable} with parameters $\theta\in\R$ and $\beta>0\in\R$, commonly denoted $X\sim Cauchy(\theta,\beta)$ if \begin{align*} f_X(x)=\frac{1}{\pi\beta[1+(\frac{x-\theta}{\beta})^2]}. \end{align*} Cauchy random variables are used primarily for theoretical purposes, the key point being that the values $E[X]$ and $Var[X]$ are undefined for Cauchy random variables.