Cauchy distribution

The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution, Cauchy–Lorentz distribution, Lorentz function, or Breit–Wigner distribution. The Cauchy distribution is the distribution of the -intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.
The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined. The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no moment generating function.
In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.
It is one of the few stable distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

Definitions

Here are the most important constructions.

Rotational symmetry

If one stands in front of a line and kicks a ball at a uniformly distributed random angle towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution.
For example, consider a point at in the x-y plane, and select a line passing through the point, with its direction chosen uniformly at random. The intersection of the line with the x-axis follows a Cauchy distribution with location and scale.
This definition gives a simple way to sample from the standard Cauchy distribution. Let be a sample from a uniform distribution from, then we can generate a sample, from the standard Cauchy distribution using
When and are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio has the standard Cauchy distribution.
More generally, if is a rotationally symmetric distribution on the plane, then the ratio has the standard Cauchy distribution.

Probability density function (PDF)

The Cauchy distribution is the probability distribution with the following probability density function
where is the location parameter, specifying the location of the peak of the distribution, and is the scale parameter which specifies the half-width at half-maximum, alternatively is full width at half maximum. is also equal to half the interquartile range and is sometimes called the probable error. This function is also known as a Lorentzian function, and an example of a nascent delta function, and therefore approaches a Dirac delta function in the limit as. Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining this Dirac delta function.

Properties of PDF

The maximum value or amplitude of the Cauchy PDF is, located at.
It is sometimes convenient to express the PDF in terms of the complex parameter
The special case when and is called the standard Cauchy distribution with the probability density function
In physics, a three-parameter Lorentzian function is often used:
where is the height of the peak. The three-parameter Lorentzian function indicated is not, in general, a probability density function, since it does not integrate to 1, except in the special case where

Cumulative distribution function (CDF)

The Cauchy distribution is the probability distribution with the following cumulative distribution function :
and the quantile function of the Cauchy distribution is
It follows that the first and third quartiles are, and hence the interquartile range is.
For the standard distribution, the cumulative distribution function simplifies to arctangent function :

Other constructions

The standard Cauchy distribution is the Student's t-distribution with one degree of freedom, and so it may be constructed by any method that constructs the Student's t-distribution.
If is a positive-semidefinite covariance matrix with strictly positive diagonal entries, then for independent and identically distributed and any random -vector independent of and such that and it holds that

Properties

The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to.
The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution.
The family of Cauchy-distributed random variables is closed under linear fractional transformations with real coefficients. In this connection, see also McCullagh's parametrization of the Cauchy distributions.

Sum of Cauchy-distributed random variables

If are an IID sample from the standard Cauchy distribution, then their sample mean is also standard Cauchy distributed. In particular, the average does not converge to the mean, and so the standard Cauchy distribution does not follow the law of large numbers.
This can be proved by repeated integration with the PDF, or more conveniently, by using the characteristic function of the standard Cauchy distribution :With this, we have, and so has a standard Cauchy distribution.
More generally, if are independent and Cauchy distributed with location parameters and scales, and are real numbers, then is Cauchy distributed with location and scale. We see that there is no law of large numbers for any weighted sum of independent Cauchy distributions.
This shows that the condition of finite variance in the central limit theorem cannot be dropped. It is also an example of a more generalized version of the central limit theorem that is characteristic of all stable distributions, of which the Cauchy distribution is a special case.

Central limit theorem

If are an IID sample with PDF such that is finite, but nonzero, then converges in distribution to a Cauchy distribution with scale.

Characteristic function

Let denote a Cauchy distributed random variable. The characteristic function of the Cauchy distribution is given by
which is just the Fourier transform of the probability density. The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform:
The nth moment of a distribution is the nth derivative of the characteristic function evaluated at. Observe that the characteristic function is not differentiable at the origin: this corresponds to the fact that the Cauchy distribution does not have well-defined moments higher than the zeroth moment.

Kullback–Leibler divergence

The Kullback–Leibler divergence between two Cauchy distributions has the following symmetric closed-form formula:
Any f-divergence between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence.
Closed-form expression for the total variation, Jensen–Shannon divergence, Hellinger distance, etc. are available.

Entropy

The entropy of the Cauchy distribution is given by:
The derivative of the quantile function, the quantile density function, for the Cauchy distribution is:
The differential entropy of a distribution can be defined in terms of its quantile density, specifically:
The Cauchy distribution is the maximum entropy probability distribution for a random variate for which

Moments

The Cauchy distribution is usually used as an illustrative counterexample in elementary probability courses, as a distribution with no well-defined moments.

Sample moments

If we take an IID sample from the standard Cauchy distribution, then the sequence of their sample mean is, which also has the standard Cauchy distribution. Consequently, no matter how many terms we take, the sample average does not converge.
Similarly, the sample variance also does not converge.
A typical trajectory of looks like long periods of slow convergence to zero, punctuated by large jumps away from zero, but never getting too far away. A typical trajectory of looks similar, but the jumps accumulate faster than the decay, diverging to infinity. These two kinds of trajectories are plotted in the figure.
Moments of sample lower than order 1 would converge to zero. Moments of sample higher than order 2 would diverge to infinity even faster than sample variance.

Mean

If a probability distribution has a density function, then the mean, if it exists, is given by
We may evaluate this two-sided improper integral by computing the sum of two one-sided improper integrals. That is,
for an arbitrary real number.
For the integral to exist, at least one of the terms in this sum should be finite, or both should be infinite and have the same sign. But in the case of the Cauchy distribution, both the terms in this sum are infinite and have opposite sign. Hence is undefined, and thus so is the mean. When the mean of a probability distribution function is undefined, no one can compute a reliable average over the experimental data points, regardless of the sample's size.
Note that the Cauchy principal value of the mean of the Cauchy distribution is
which is zero. On the other hand, the related integral
is not zero, as can be seen by computing the integral. This again shows that the mean cannot exist.
Various results in probability theory about expected values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.

Smaller moments

The absolute moments for are defined.
For we have

Higher moments

The Cauchy distribution does not have finite moments of any order. Some of the higher raw moments do exist and have a value of infinity, for example, the raw second moment:
By re-arranging the formula, one can see that the second moment is essentially the infinite integral of a constant. Higher even-powered raw moments will also evaluate to infinity. Odd-powered raw moments, however, are undefined, which is distinctly different from existing with the value of infinity. The odd-powered raw moments are undefined because their values are essentially equivalent to since the two halves of the integral both diverge and have opposite signs. The first raw moment is the mean, which, being odd, does not exist. This in turn means that all of the central moments and standardized moments are undefined since they are all based on the mean. The variance—which is the second central moment—is likewise non-existent.
The results for higher moments follow from Hölder's inequality, which implies that higher moments diverge if lower ones do.