# 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`

## Sources

- 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions

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