Cauchy distribution

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

in the x-y plane, and select a line passing through the point, with its direction (angle with the

The intersection of the line with the x-axis follows a Cauchy distribution with location

are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio

positive-semidefinite covariance matrix with strictly positive diagonal entries, then for independent and identically distributed

In addition, the family of Cauchy-distributed random variables is closed under linear fractional transformations with real coefficients.

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 (see below):

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.

The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform:

The Kullback–Leibler divergence between two Cauchy distributions has the following symmetric closed-form formula:[11]

Any f-divergence between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence.

The differential entropy of a distribution can be defined in terms of its quantile density,[13] specifically:

But in the case of the Cauchy distribution, both the terms in this sum (2) are infinite and have opposite sign.

[15] When the mean of a probability distribution function (PDF) is undefined, no one can compute a reliable average over the experimental data points, regardless of the sample's size.

Various results in probability theory about expected values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.

Odd-powered raw moments, however, are undefined, which is distinctly different from existing with the value of infinity.

The variance—which is the second central moment—is likewise non-existent (despite the fact that the raw second moment exists with the value infinity).

[16] Because the parameters of the Cauchy distribution do not correspond to a mean and variance, attempting to estimate the parameters of the Cauchy distribution by using a sample mean and a sample variance will not succeed.

Similarly, calculating the sample variance will result in values that grow larger as more observations are taken.

Other, more precise and robust methods have been developed [20][21] For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for

[22][23] However, because of the fat tails of the Cauchy distribution, the efficiency of the estimator decreases if more than 24% of the sample is used.

[24] Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.

[23][27] The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of

The shape can be estimated using the median of absolute values, since for location 0 Cauchy variables

[citation needed] A function with the form of the density function of the Cauchy distribution was studied geometrically by Fermat in 1659, and later was known as the witch of Agnesi, after Maria Gaetana Agnesi included it as an example in her 1748 calculus textbook.

[36] Poisson noted that if the mean of observations following such a distribution were taken, the standard deviation did not converge to any finite number.

As such, Laplace's use of the central limit theorem with such a distribution was inappropriate, as it assumed a finite mean and variance.

Despite this, Poisson did not regard the issue as important, in contrast to Bienaymé, who was to engage Cauchy in a long dispute over the matter.

A typical trajectory of sample means 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 sample variances looks similar, but the jumps accumulate faster than the decay, diverging to infinity.
Fitted cumulative Cauchy distribution to maximum one-day rainfalls using CumFreq , see also distribution fitting [ 31 ]
Estimating the mean and standard deviation through a sample from a Cauchy distribution (bottom) does not converge as the size of the sample grows, as in the normal distribution (top). There can be arbitrarily large jumps in the estimates, as seen in the graphs on the bottom. (Click to expand)