Distribution name

Cauchy distribution

Common notation

Parameters

 = location parameter

 = scale parameter (

Domain

Probability density function

Cumulative distribution function

Mean

Does not exist

Variance

Does not exist

Skewness

Does not exist

(Excess) kurtosis

Does not exist

Characteristic function

Other comments

The quantile function of the Cauchy distribution is:

Its median is thus .

The Cauchy distribution is a special case of the stable (more precisely the sum stable) distribution family.

The special case of the Cauchy distribution when  and  is called the standard Cauchy distribution. It coincides with the Student’s t distribution with one degree of freedom. It has a probability density function of .

If  and  are independent random variables then  and this can be used to generate random variates.

The Cauchy distribution is also known as the Cauchy-Lorentz or Lorentz distribution (especially amongst physicists).