Cauchy distribution

The Cauchy distribution is a probability distribution with probability density function:

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

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:

<math> f(x) = \frac{1}{\pi (1 + x^2)} </math>

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 normal random variables with standard normal distributions, then the ratio U/V has the standard Cauchy distribution.

The Cauchy distribution is sometimes called the Lorentz distribution