Cauchy Distribution

The Cauchy distribution, also known as the Lorentzian distribution, is a continuous probability distribution that serves as an important example of a stable distribution without finite moments. Named after Augustin Louis Cauchy, it is notable for its unique properties: undefined mean, undefined variance, and heavy tails that decay as a power law. It belongs to the family of stable distributions and appears naturally in physical and mathematical applications.

The Cauchy distribution is characterized by two parameters: the location parameter x₀ (which can take any real value) and the scale parameter γ (which must be strictly positive). Its probability density function (PDF) is:

This distribution is defined for all real values x, with x₀ determining the peak location and γ controlling the spread. The standard Cauchy distribution arises when x₀ = 0 and γ = 1.

The cumulative distribution function (CDF) is given by:

Like the PDF, the CDF is defined over the entire real line and provides the probability that a Cauchy-distributed random variable takes on a value less than or equal to x.

The Cauchy distribution appears in various physical and statistical contexts:

• The Cauchy distribution describes spectral line broadening in physics, particularly in the form of the Lorentzian line shape.

• It serves as a fundamental model in the mathematical description of resonance behavior in physical systems.

• In robust statistics, it is used as a prototype heavy-tailed distribution for modeling data with extreme values.

• The distribution plays a crucial role in understanding limit distributions of certain random processes and statistical phenomena.

The Cauchy distribution is unique among common probability distributions due to its lack of finite moments:

Additional notable properties of the Cauchy distribution include:

• It is symmetric about its location parameter x₀, which is also the mode.

• The distribution is scale invariant: if X is Cauchy distributed, then aX + b (for real a ≠ 0, b) is also Cauchy distributed.

• The distribution is its own conjugate prior in Bayesian inference.

The Cauchy distribution has several important relationships with other distributions:

• It is a special case of the Student's t-distribution with one degree of freedom

• It can be generated as the ratio of two independent standard normal random variables

• It is a stable distribution with stability parameter $\alpha =1$

• The sum of independent Cauchy random variables is also Cauchy distributed

The Cauchy distribution is implemented by the CauchyDistribution class. It has one constructor which takes the location and scale parameters as arguments. The following constructs a Cauchy distribution with location parameter 0 and scale parameter 3.2:

var cauchy = new CauchyDistribution(3.2);

Dim cauchy = New CauchyDistribution(3.2)

No code example is currently available or this language may not be supported.

let cauchy = CauchyDistribution(3.2)

The CauchyDistribution class has two specific properties: LocationParameter and ScaleParameter.

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 Pcg32();
double sample = CauchyDistribution.Sample(random, 3.2);

Dim random = New Pcg32()
Dim sample = CauchyDistribution.Sample(random, 3.2)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = CauchyDistribution.Sample(random, 3.2)

For more information on the Cauchy distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 16. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 10. Wiley Series in Probability and Statistics.