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:


 * $\displaystyle \map {f_X} x = \frac 1 {\pi \gamma \paren {1 + \paren {\frac {x - x_0} \gamma}^2} }$

for some $\gamma > 0$.

This is written:


 * $X \sim \operatorname {Cauchy} \paren {x_0, \gamma}$