# Cauchy: Create a Cauchy distribution In distributions3: Probability Distributions as S3 Objects

## Description

Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.

## Usage

 `1` ```Cauchy(location = 0, scale = 1) ```

## Arguments

 `location` The location parameter. Can be any real number. Defaults to `0`. `scale` The scale parameter. Must be greater than zero (?). Defaults to `1`.

## Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

In the following, let X be a Cauchy variable with mean `location =` x_0 and `scale` = γ.

Support: R, the set of all real numbers

Mean: Undefined.

Variance: Undefined.

Probability density function (p.d.f):

f(x) = 1 / (π γ (1 + ((x - x_0) / γ)^2)

Cumulative distribution function (c.d.f):

F(t) = arctan((t - x_0) / γ) / π + 1/2

Moment generating function (m.g.f):

Does not exist.

## Value

A `Cauchy` object.

Other continuous distributions: `Beta`, `ChiSquare`, `Exponential`, `FisherF`, `Gamma`, `LogNormal`, `Logistic`, `Normal`, `StudentsT`, `Tukey`, `Uniform`, `Weibull`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```set.seed(27) X <- Cauchy(10, 0.2) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 2) quantile(X, 0.7) cdf(X, quantile(X, 0.7)) quantile(X, cdf(X, 7)) ```