Cauchy random variable

$X$ is a Cauchy random variable with parameters $\theta \in ℝ$ and $\beta >0\in ℝ$, commonly denoted $X\sim Cauchy(\theta ,\beta )$ if

$f_X(x)={\displaystyle \frac{1}{\pi \beta [1+{(\frac{x-\theta }{\beta })}^2]}}.$

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.