Cauchy distribution

Story

The intercept on the x-axis of a beam of light coming from the point \((\mu, \sigma)\) is Cauchy distributed. This story is popular in physics, but is not particularly useful. You can think of it as a peaked distribution with enormously heavy tails.

Parameters

The Cauchy distribution is peaked, and its peak is located at \(\mu\), its location parameter, which may take on any real value. The peak’s width is dictated by a positive scale parameter \(\sigma\).

Support

The Cauchy distribution is supported on the set of real numbers.

Probability density function

\[\begin{align} f(y;\mu,\sigma) = \frac{1}{\pi \sigma}\,\frac{1}{1 + (y-\mu)^2/\sigma^2}. \end{align}\]

Cumulative distribution function

\[\begin{align} F(y;\mu,\sigma) = \frac{1}{2} + \frac{1}{\pi}\,\arctan\left(\frac{y - \mu}{\sigma}\right). \end{align}\]

Moments

Mean: Undefined

Variance: Undefined

Usage

Package

Syntax

NumPy

mu + sigma * rng.standard_cauchy()

SciPy

scipy.stats.cauchy(mu, sigma)

Distributions.jl

``Cauchy(mu, sigma) ``

Stan

cauchy(mu, sigma)

Notes

  • The numpy.random module only has the Standard Cauchy distribution (\(\mu=0\) and \(\sigma=1\)), but you can draw out of a Cauchy distribution using the transformation shown in the NumPy usage above.

  • The Cauchy distribution has extremely heavy tails, so heavy that no moments are defined.

PDF and CDF plots