# 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:

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

for some $\gamma > 0$.

This is written:

- $X \sim \Cauchy {x_0} \gamma$

## Also see

- Results about
**the Cauchy distribution**can be found here.

## Source of Name

This entry was named for Augustin Louis Cauchy.

## Technical Note

The $\LaTeX$ code for \(\Cauchy {x_0} {\gamma}\) is `\Cauchy {x_0} {\gamma}`

.

When either of the arguments is a single character, it is usual to omit the braces:

`\Cauchy {x_0} \gamma`

## Sources

- Weisstein, Eric W. "Cauchy Distribution." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html