is a Cauchy random variable with parameters and , commonly denoted if
Cauchy random variables are used primarily for theoretical purposes, the key point being that the values and are undefined for Cauchy random variables.
![$\displaystyle f_X(x)=\frac{1}{\pi\beta[1+(\frac{x-\theta}{\beta})^2]}.$](http://images.planetmath.org:8080/cache/objects/531/l2h/img5.png "$\displaystyle f_X(x)=\frac{1}{\pi\beta[1+(\frac{x-\theta}{\beta})^2]}.$")
![$ E[X]$](http://images.planetmath.org:8080/cache/objects/531/l2h/img6.png "$ E[X]$"){kind=small}
![$ Var[X]$](http://images.planetmath.org:8080/cache/objects/531/l2h/img7.png "$ Var[X]$"){kind=small}