Cauchy distribution

LorentzianCauchyLorentzian distributionLorentzian functionLorentzian profileCauchy–Lorentz distributionCauchy-Lorentz distributionLorentzian or Cauchy functionMultivariate CauchyCauchy cumulative distribution function
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.wikipedia
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Ratio distribution

ratio distributionsComplex normal ratio distributionGaussian ratio distribution
It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.
An example is the Cauchy distribution (also called the normal ratio distribution), which comes about as the ratio of two normally distributed variables with zero mean.

Normal distribution

normally distributedGaussian distributionnormal
It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.
However, many other distributions are bell-shaped (such as the Cauchy, Student's t-, and logistic distributions).

Stable distribution

stable distributionsGeneralized Central Limit TheoremLévy alpha-stable distribution
It is one of the few distributions that is stable and has a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.
Stable distributions have 0 < α ≤ 2, with the upper bound corresponding to the normal distribution, and α = 1 to the Cauchy distribution.

Variance

sample variancepopulation variancevariability
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 is an example of a distribution which has no mean, variance or higher moments defined.
If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either.

Hendrik Lorentz

Hendrik Antoon LorentzLorentzH. A. Lorentz
It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution.
Lorentz' name is now associated with the Lorentz-Lorenz formula, the Lorentz force, the Lorentzian distribution, and the Lorentz transformation.

Witch of Agnesi

Versiera
Functions with the form of the density function of the Cauchy distribution were studied by mathematicians in the 17th century, but in a different context and under the title of the witch of Agnesi.
As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory.

Expected value

expectationexpectedmean
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. Various results in probability theory about expected values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.
An example of such a random variable is one with the Cauchy distribution, due to its large "tails".

Moment-generating function

moment generating functionCalculations of momentsgenerating functions
The Cauchy distribution has no moment generating function.

Pathological (mathematics)

well-behavedpathologicalbadly behaved
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.
For example, in statistics, the Cauchy distribution does not satisfy the central limit theorem, even though its symmetric bell-shape appears similar to many distributions which do; it fails the requirement to have a mean and standard deviation which exist and that are finite.

Dirac delta function

Dirac deltadelta functionimpulse
Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function.
An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy.

Mean

mean valueaveragepopulation mean
The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined.
Not every probability distribution has a defined mean; see the Cauchy distribution for an example.

Median

averagesample medianmedian-unbiased estimator
Its mode and median are well defined and are both equal to x_0.

McCullagh's parametrization of the Cauchy distributions

In this connection, see also McCullagh's parametrization of the Cauchy distributions.
In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function (pdf) is

Probable error

\gamma is also equal to half the interquartile range and is sometimes called the probable error.
One such use of the term probable error in this sense is as the name for the scale parameter of the Cauchy distribution, which does not have a standard deviation.

Scale parameter

scalerate parameterestimation
where x_0 is the location parameter, specifying the location of the peak of the distribution, and \gamma is the scale parameter which specifies the half-width at half-maximum (HWHM), alternatively 2\gamma is full width at half maximum (FWHM).

Full width at half maximum

FWHMhalf-widthbeamwidth
where x_0 is the location parameter, specifying the location of the peak of the distribution, and \gamma is the scale parameter which specifies the half-width at half-maximum (HWHM), alternatively 2\gamma is full width at half maximum (FWHM).
For example, a Lorentzian/Cauchy distribution of height 1⁄γ can be defined by

Stability (probability)

Stabilitystable
It is also a strictly stable distribution.
Important special cases of stable distributions are the normal distribution, the Cauchy distribution and the Lévy distribution.

Infinite divisibility (probability)

infinitely divisibleinfinite divisibilityinfinitely divisible distribution
The Cauchy distribution is an infinitely divisible probability distribution.
The Poisson distribution, the negative binomial distribution, the Gamma distribution and the degenerate distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family.

Central limit theorem

Lyapunov's central limit theoremlimit theoremscentral limit
As such, Laplace's use of the Central Limit Theorem with such a distribution was inappropriate, as it assumed a finite mean and variance.
Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined.

Law of large numbers

strong law of large numbersweak law of large numbersBernoulli's Golden Theorem
Various results in probability theory about expected values, such as the strong law of large numbers, fail to hold for the Cauchy distribution.
For instance, the average of the results from Cauchy distribution or some Pareto distribution (α

Characteristic function (probability theory)

characteristic functioncharacteristic functionscharacteristic function:
To see that this is true, compute the characteristic function of the sample mean:
For example, suppose X has a standard Cauchy distribution.

Differential entropy

differential entropiesdifferential Shannon informationentropy
The differential entropy of a distribution can be defined in terms of its quantile density, specifically:

Augustin-Louis Cauchy

CauchyAugustin Louis CauchyAugustin Cauchy
Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function. The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.

Probability density function

probability densitydensity functiondensity
The Cauchy distribution has the probability density function (PDF)
This is the density of a standard Cauchy distribution.

Irénée-Jules Bienaymé

BienayméBienaymé, Irénée-JulesI. J. Bienaymé
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.
Cauchy developed the Cauchy distribution to show a case where the method of ordinary least squares resulted in a perfectly inefficient estimator.