The Cauchy Distribution

The Cauchy distribution can be used to model the distribution of horizontal distances at which a line segment tilted at a random angle cuts a line.

The Cauchy distribution has one scale parameter. The probability density function is:

Probability density of the Cauchy distribution.

The Cauchy distribution looks similar to a normal distribution, but has a much heavier tail. It can be used to study hypothesis tests that assume a normal distribution. How well the tests perform on data from a Cauchy distribution gives a good indication of the sensitivity of the test to heavy-tail departures from normality.

The mean and standard deviation of the Cauchy distribution are undefined. This means that no amount of data points will yield a more accurate or reliable estimate of the mean and standard deviation than does a single point.

The Cauchy distribution is sometimes called the Lorentzian distribution.

The Cauchy distribution is implemented by the CauchyDistribution class. It has one constructor which takes the scale parameter as its only argument. The following constructs a beta distribution with scale parameter 3.2:

var cauchy = new CauchyDistribution(3.2);

The CauchyDistribution class has one specific property, ScaleParameter, that returns the scale parameter of the distribution.

CauchyDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new MersenneTwister();
double sample = CauchyDistribution.Sample(random, 3.2);

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..