Cauchy distribution

The Cauchy distribution is a probability distribution with probability density function

$f(x)={\frac {1}{s\pi [1+((x-t)/s)^{2}]}}$

where t is the location parameter and s is the scale parameter. The special case when t = 0 and s = 1 is called the standard Cauchy distribution with the probability density function

$f(x)={\frac {1}{\pi (1+x^{2})}}.$

The Cauchy distribution is often cited as an example of a distribution which has no mean, variance or higher moments defined, although its mode and median are well defined and both zero.

When U and V are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio U/V has the standard Cauchy distribution.

If X₁, ..., X_n are independent random variables, each with a standard Cauchy distribution, the the sample mean (X₁ + ... + X_n)/n has the same standard Cauchy distribution. This example serves to show that the hypothesis of finite variance in the central limit theorem cannot be dropped (although it can be replaced with other, in some cases weaker, assumptions). To see that this is true, compute the characteristic function

$\varphi (t)=E\left(e^{it{\overline {X}}}\right)$

where ${\overline {X}}$ is the sample mean.

The Cauchy distribution is the Student's t-distribution with just one degree of freedom.

The Cauchy distribution is sometimes called the Lorentz distribution.