Cauchy Distribution

The Cauchy distribution, named after the mathematician Augustin-Louis Cauchy, is a continuous probability distribution with significant applications in statistics, particularly as an example of a "pathological" distribution. It is also known as the Lorentz distribution, or Lorentzian function, especially in physics, and serves as the distribution of the ratio of two independent normally distributed random variables with zero mean.

Characteristics

Probability Density Function

The probability density function (PDF) of a Cauchy distribution is defined as:

[ f(x; x_0, \gamma) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]} ]

where ( x_0 ) is the location parameter and ( \gamma ) is the scale parameter. The peak of the distribution is at ( x_0 ).

Cumulative Distribution Function

The cumulative distribution function (CDF) simplifies to:

[ F(x; x_0, \gamma) = \frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right) + \frac{1}{2} ]

Undefined Moments

A key characteristic of the Cauchy distribution is that its expected value and variance are undefined. This feature is due to the distribution's "fat tails," a hallmark of a fat-tailed distribution. The standard Cauchy distribution is equivalent to the Student's t-distribution with one degree of freedom, highlighting its unique properties.

Mathematical Properties

Stability

The Cauchy distribution is an example of a strictly stable distribution, which means that a linear combination of two independent Cauchy random variables results in another Cauchy random variable. This property is shared with the family of stable distributions.

Infinite Divisibility

It is also an infinitely divisible probability distribution. This means that for any natural number ( n ), there exist ( n ) independent identically distributed random variables whose sum follows a Cauchy distribution.

Relation to Other Distributions

The Cauchy distribution is closely related to other distributions, notably:

Wrapped Cauchy Distribution: This is a wrapped probability distribution that arises by "wrapping" the Cauchy distribution around the unit circle.
Log-Cauchy Distribution: This is a probability distribution of a random variable whose logarithm has a Cauchy distribution.
Cauchy Principal Value: Used in complex analysis to assign values to certain improper integrals, which would otherwise be undefined.

Applications

The Cauchy distribution is used in various fields, including physics, where it is known as the Lorentz distribution and is used to describe resonance behavior. It often appears in problems related to spectroscopy and other applications involving resonant systems.