The Cauchy distribution
is a probability distribution
with probability density function
is the location parameter
is the scale parameter
. The special case when t
= 0 and s
= 1 is called
the standard Cauchy distribution
with the probability density function
The Cauchy distribution is often cited as an example of a distribution which has no mean
or higher moments
defined, although its mode
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, ..., Xn are independent random variables, each with a standard Cauchy distribution, the the sample mean (X1 + ... + Xn)/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
where 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.