Definition:Cauchy Distribution

Definition
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:


 * $\map {f_X} x = \dfrac 1 {\pi \lambda \paren {1 + \paren {\frac {x - \gamma} \lambda }^2} }$

for:
 * $\lambda \in \R_{>0}$
 * $\gamma \in \R$

This is written:


 * $X \sim \Cauchy \gamma \lambda$