# 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:

$X \sim \Cauchy \gamma \lambda$

### Scale Parameter

Let $X$ be a continuous random variable with a Cauchy distribution:

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

for:

$\lambda \in \R_{>0}$
$\gamma \in \R$

The parameter $\lambda$ is referred to as the scale parameter of $X$.

### Standard

The standard Cauchy distribution is the Cauchy distribution with $\lambda = 1$ and $\gamma = 0$:

$\map {f_X} x = \dfrac 1 {\pi \paren {1 + x^2} }$

## Also presented as

The Cauchy Distribution of $X$ can be found expressed in the form:

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

## Also defined as

Some sources define the Cauchy distribution of $X$ as the standard Cauchy distribution:

$\map {f_X} x = \dfrac 1 {\pi \paren {1 + x^2} }$

which is obtained from the full form by setting:

$\lambda = 1$
$\gamma = 0$

Some sources give it as:

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

which is obtained from the full form by setting $\lambda = 1$.

## Notation

The notation used to denote a Cauchy distribution is generally consistent, except for the parameter labels.

Some sources use $\theta$ for $\gamma$.

MathWorld uses $m$ and $b$ for $\lambda$ and $\gamma$.

All notation is perfectly good, as long as it is clear what the parameters are.

## 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 $\Cauchy {\gamma} {\lambda}$ is \Cauchy {\gamma} {\lambda} .

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

\Cauchy \gamma \lambda