Cauchy Distribution

The Cauchy distribution is a symmetrical, and to use a technical term, heavy tailed.  Heavy tailed means that a high proportion of the population is comprised of extreme values.

There is no analytical definition of moment based properties (e.g. mean, variance etc.) thus the parameters are typically described as the location parameter and a scale factor.  The most easily derived property is the median for this reason and for consistency with the rest of the site, the parameters have been defined as the median and a scale factor.

The moment based properties derived from a set of random numbers do not provide any useful information on the properties of the distribution.

The Cauchy distribution is also known as the Lorentzian Distribution.

An application of the Cauchy distribution is in software testing where it is necessary to use datasets which contain a few extreme values which might trigger some adverse reaction.

Profile

Parameters

Parameter	Description	Characteristics
median	Also referred to as the location parameter.	A float > -∞ and < ∞
scale	A scale factor which determines the concentration of data around the mode.	A float > 0

Range

The range of the Cauchy distribution is from -∞ to +∞.

Functions

Properties

Whilst the moment based properties are not defined, a corollary of the distribution being symmetrical is that the mode and median will have the same value as the median and that skewness has the value zero.

Random Number Generation

Random number generation (referred to as r) for a Cauchy distribution can be performed by transforming a continuous uniform variable in the range 0 to 1 (referred to as u) with the distribution's inverse probability function:

r=g(u)

Using psuedo code, the function would be similar to:

r=median+scale * tan(π*u)