\begin{definition}[Definition:Cauchy Distribution]
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Img X = \R$. 
$X$ is said to have a '''Cauchy distribution''' if it has probability density function:
:$\map {f_X} x = \dfrac 1 {\pi \gamma \paren {1 + \paren {\frac {x - x_0} \gamma}^2} }$
for some $\gamma > 0$. 
This is written: 
:$X \sim \Cauchy {x_0} \gamma$
\end{definition}