Cauchy distribution

Cauchy

Probability density function Probability density function for the Cauchy distribution The purple curve is the standard Cauchy distribution
Cumulative distribution function Cumulative distribution function for the Cauchy distribution
Parameters	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0\!} location (real) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma > 0} scale (real)
Support	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle x\in (-\infty ,+\infty )\!}
PDF	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!}
CDF	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!}
Quantile	${\displaystyle x_{0}+\gamma \,\tan[\pi (p-{\tfrac {1}{2}})]}$
Mean	undefined
Median	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{0}\!}
Mode	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{0}\!}
Variance	undefined
MAD	${\displaystyle \gamma }$
Skewness	undefined
Ex. kurtosis	undefined
Entropy	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \log(4\pi \gamma )\!}
MGF	does not exist
CF	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle \exp(x_{0}\,i\,t-\gamma \,|t|)\!}
Fisher information	Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{2\gamma ^{2}}}}

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution ${\displaystyle f(x;x_{0},\gamma )}$ is the distribution of the x-intercept of a ray issuing from Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x_{0},\gamma )} 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 (but see § Explanation of undefined moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist.^[1] 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 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.

History[edit | edit source]

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 Agnesi included it as an example in her 1748 calculus textbook. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French mathematician Poisson in 1824, with Cauchy only becoming associated with it during an academic controversy in 1853.^[2] Poisson noted that if the mean of observations following such a distribution were taken, the mean error^{[further explanation needed]} did not converge to any finite number. As such, Laplace's use of the central limit theorem with such 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.

Characterisation[edit | edit source]

Probability density function[edit | edit source]

The Cauchy distribution has the probability density function (PDF)^[1][3]

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x;x_{0},\gamma )={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}={1 \over \pi \gamma }\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}}\right],}

where ${\displaystyle x_{0}}$ is the location parameter, specifying the location of the peak of the distribution, and ${\displaystyle \gamma }$ is the scale parameter which specifies the half-width at half-maximum (HWHM), alternatively Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\gamma } is full width at half maximum (FWHM). ${\displaystyle \gamma }$ is also equal to half the interquartile range and is sometimes called the probable error. 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 maximum value or amplitude of the Cauchy PDF is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{\pi \gamma }}} , located at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=x_{0}} .

It is sometimes convenient to express the PDF in terms of the complex parameter Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \psi =x_{0}+i\gamma }

${\displaystyle f(x;\psi )={\frac {1}{\pi }}\,{\textrm {Im}}\left({\frac {1}{x-\psi }}\right)={\frac {1}{\pi }}\,{\textrm {Re}}\left({\frac {-i}{x-\psi }}\right)}$

The special case when Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x_{0}=0} and ${\displaystyle \gamma =1}$ is called the standard Cauchy distribution with the probability density function^[4][5]

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x;0,1)={\frac {1}{\pi (1+x^{2})}}.\!}

In physics, a three-parameter Lorentzian function is often used:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x;x_{0},\gamma ,I)={\frac {I}{\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}=I\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{2}}\right],}

where ${\displaystyle I}$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I={\frac {1}{\pi \gamma }}.\!}

Cumulative distribution function[edit | edit source]

The cumulative distribution function of the Cauchy distribution is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x;x_{0},\gamma )={\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}}

and the quantile function (inverse cdf) of the Cauchy distribution is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q(p;x_{0},\gamma )=x_{0}+\gamma \,\tan \left[\pi \left(p-{\tfrac {1}{2}}\right)\right].}

It follows that the first and third quartiles are ${\displaystyle (x_{0}-\gamma ,x_{0}+\gamma )}$, and hence the interquartile range is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2\gamma } .

For the standard distribution, the cumulative distribution function simplifies to arctangent function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \arctan(x)} :

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x;0,1)={\frac {1}{\pi }}\arctan \left(x\right)+{\frac {1}{2}}}

Entropy[edit | edit source]

The entropy of the Cauchy distribution is given by:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}H(\gamma )&=-\int _{-\infty }^{\infty }f(x;x_{0},\gamma )\log(f(x;x_{0},\gamma ))\,dx\\[6pt]&=\log(4\pi \gamma )\end{aligned}}}

The derivative of the quantile function, the quantile density function, for the Cauchy distribution is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Q'(p;\gamma )=\gamma \,\pi \,{\sec }^{2}\left[\pi \left(p-{\tfrac {1}{2}}\right)\right].\!}

The differential entropy of a distribution can be defined in terms of its quantile density,^[6] specifically:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H(\gamma )=\int _{0}^{1}\log \,(Q'(p;\gamma ))\,\mathrm {d} p=\log(4\pi \gamma )}

The Cauchy distribution is the maximum entropy probability distribution for a random variate ${\displaystyle X}$ for which

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {E} [\log(1+(X-x_{0})^{2}/\gamma ^{2})]=\log 4}

or, alternatively, for a random variate ${\displaystyle X}$ for which

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {E} [\log(1+(X-x_{0})^{2})]=2\log(1+\gamma ).}

In its standard form, it is the maximum entropy probability distribution for a random variate ${\displaystyle X}$ for which^[7]

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {E} \!\left[\ln(1+X^{2})\right]=\ln 4.}

Kullback-Leibler divergence[edit | edit source]

The Kullback-Leibler divergence between two Cauchy distributions has the following symmetric closed-form formula:^[8]

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {KL} \left(p_{x_{0,1},\gamma _{1}}:p_{x_{0,2},\gamma _{2}}\right)=\log {\frac {\left(\gamma _{1}+\gamma _{2}\right)^{2}+\left(x_{0,1}-x_{0,2}\right)^{2}}{4\gamma _{1}\gamma _{2}}}.}

Any f-divergence between two Cauchy distributions is symmetric and can be expressed as a function of the chi-squared divergence.^[9] Closed-form expression for the total variation, Jensen–Shannon divergence, Hellinger distance, etc are available.

Properties[edit | edit source]

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 ${\displaystyle x_{0}}$.

When ${\displaystyle U}$ and ${\displaystyle V}$ are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio ${\displaystyle U/V}$ has the standard Cauchy distribution.

If ${\displaystyle \Sigma }$ is a Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p\times p} positive-semidefinite covariance matrix with strictly positive diagonal entries, then for independent and identically distributed Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X,Y\sim N(0,\Sigma )} and any random ${\displaystyle p}$-vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} independent of ${\displaystyle X}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1+\cdots+w_p=1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_i\geq 0, i=1,\ldots,p,} (defining a categorical distribution) it holds that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j=1}^p w_j\frac{X_j}{Y_j}\sim\mathrm{Cauchy}(0,1).} ^[10]

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1, \ldots, X_n} are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_1 + \cdots + X_n)/n} has the same standard Cauchy distribution. To see that this is true, compute the characteristic function of the sample mean:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{\overline{X}}(t) = \mathrm{E}\left[e^{i\overline{X}t}\right]}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{X}} is the sample mean. This example serves to show 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.

The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution.^[11]

The standard Cauchy distribution coincides with the Student's t-distribution with one degree of freedom.

Like all stable distributions, the location-scale family to which the Cauchy distribution belongs is closed under linear transformations with real coefficients. In addition, the Cauchy distribution is closed under linear fractional transformations with real coefficients.^[12] In this connection, see also McCullagh's parametrization of the Cauchy distributions.

Characteristic function[edit | edit source]

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} denote a Cauchy distributed random variable. The characteristic function of the Cauchy distribution is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_X(t) = \operatorname{E}\left[e^{iXt} \right ] =\int_{-\infty}^\infty f(x;x_0,\gamma)e^{ixt}\,dx = e^{ix_0t - \gamma |t|}.}

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x; x_0,\gamma) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi_X(t;x_0,\gamma)e^{-ixt} \, dt \!}

The nth moment of a distribution is the nth derivative of the characteristic function evaluated at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0} . 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.

Comparison with the normal distribution[edit | edit source]

Compared to the normal distribution, the Cauchy density function has a higher peak and lower tails. An example is shown in the two figures added here

The figure to the left shows the Cauchy probability density function fitted to an observed histogram. The peak of the function is higher than the peak of the histogram while the tails are lower than those of the histogram.
The figure to the right shows the normal probability density function fitted to the same observed histogram. The peak of the function is lower than the peak of the histogram.
This illustates the above statement.

Explanation of undefined moments[edit | edit source]

Mean[edit | edit source]

If a probability distribution has a density function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} , 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} .

For the integral to exist (even as an infinite value), 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 (2) are infinite and have opposite sign. Hence (1) is undefined, and thus so is the mean.^[14]

Note that the Cauchy principal value of the mean of the Cauchy distribution is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{a\to\infty}\int_{-a}^a x f(x)\,dx } which is zero. On the other hand, the related integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{a\to\infty}\int_{-2a}^a x f(x)\,dx } is not zero, as can be seen by computing the integral. This again shows that the mean (1) 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.^[14]

Smaller moments[edit | edit source]

The absolute moments for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in(-1,1)} are defined. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\sim\mathrm{Cauchy}(0,\gamma)} we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E}[|X|^p] = \gamma^p \mathrm{sec}(\pi p/2).}

Higher moments[edit | edit source]

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \operatorname{E}[X^2] & \propto \int_{-\infty}^\infty \frac{x^2}{1+x^2}\,dx = \int_{-\infty}^\infty 1 - \frac{1}{1+x^2}\,dx \\[8pt] & = \int_{-\infty}^\infty dx - \int_{-\infty}^\infty \frac{1}{1+x^2}\,dx = \int_{-\infty}^\infty dx-\pi = \infty. \end{align} }

By re-arranging the formula, one can see that the second moment is essentially the infinite integral of a constant (here 1). 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty - \infty} 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. (See also the discussion above about this.) 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 (despite the fact that the raw second moment exists with the value infinity).

The results for higher moments follow from Hölder's inequality, which implies that higher moments (or halves of moments) diverge if lower ones do.

Moments of truncated distributions[edit | edit source]

Consider the truncated distribution defined by restricting the standard Cauchy distribution to the interval [−10¹⁰⁰, 10¹⁰⁰]. Such a truncated distribution has all moments (and the central limit theorem applies for i.i.d. observations from it); yet for almost all practical purposes it behaves like a Cauchy distribution.^[15]

Estimation of parameters[edit | edit source]

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.^[16] For example, if an i.i.d. sample of size n is taken from a Cauchy distribution, one may calculate the sample mean as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{x}=\frac 1 n \sum_{i=1}^n x_i}

Although the sample values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} will be concentrated about the central value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} , the sample mean will become increasingly variable as more observations are taken, because of the increased probability of encountering sample points with a large absolute value. In fact, the distribution of the sample mean will be equal to the distribution of the observations themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} than any single observation from the sample. Similarly, calculating the sample variance will result in values that grow larger as more observations are taken.

Therefore, more robust means of estimating the central value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and the scaling parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} are needed. One simple method is to take the median value of the sample as an estimator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and half the sample interquartile range as an estimator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} . Other, more precise and robust methods have been developed ^[17][18] For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} that is more efficient than using either the sample median or the full sample mean.^[19][20] 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.^[19][20]

Maximum likelihood can also be used to estimate the parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} . However, this tends to be complicated by the fact that this requires finding the roots of a high degree polynomial, and there can be multiple roots that represent local maxima.^[21] Also, while the maximum likelihood estimator is asymptotically efficient, it is relatively inefficient for small samples.^[22][23] The log-likelihood function for the Cauchy distribution for sample size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat\ell(x_1,\dotsc,x_n \mid \!x_0,\gamma ) = - n \log (\gamma \pi) - \sum_{i=1}^n \log \left(1 + \left(\frac{x_i - x_0}{\gamma}\right)^2\right)}

Maximizing the log likelihood function with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} by taking the first derivative produces the following system of equations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \ell}{d x_{0}} = \sum_{i=1}^n \frac{2(x_i - x_0)}{\gamma^2 + \left(x_i - \!x_0\right)^2} =0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \ell}{d \gamma} = \sum_{i=1}^n \frac{2\left(x_i - x_0\right)^2}{\gamma (\gamma^2 + \left(x_i - x_0\right)^2)} - \frac{n}{\gamma} = 0}

Note that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n \frac{\left(x_i - x_0\right)^2}{\gamma^2 + \left(x_i - x_0\right)^2} }

is a monotone function in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} and that the solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} must satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \min |x_i-x_0|\le \gamma\le \max |x_i-x_0|. }

Solving just for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} requires solving a polynomial of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n-1} ,^[21] and solving just for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \,\!\gamma} requires solving a polynomial of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} . Therefore, whether solving for one parameter or for both parameters simultaneously, a numerical solution on a computer is typically required. The benefit of maximum likelihood estimation is asymptotic efficiency; estimating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} using the sample median is only about 81% as asymptotically efficient as estimating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} by maximum likelihood.^[20][24] The truncated sample mean using the middle 24% order statistics is about 88% as asymptotically efficient an estimator of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} as the maximum likelihood estimate.^[20] When Newton's method is used to find the solution for the maximum likelihood estimate, the middle 24% order statistics can be used as an initial solution for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} .

The shape can be estimated using the median of absolute values, since for location 0 Cauchy variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\sim\mathrm{Cauchy}(0,\gamma)} , the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{median}(|X|) = \gamma} the shape parameter.

Multivariate Cauchy distribution[edit | edit source]

A random vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=(X_1, \ldots, X_k)^T} is said to have the multivariate Cauchy distribution if every linear combination of its components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y=a_1X_1+ \cdots + a_kX_k} has a Cauchy distribution. That is, for any constant vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in \mathbb R^k} , the random variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y=a^TX} should have a univariate Cauchy distribution.^[25] The characteristic function of a multivariate Cauchy distribution is given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_X(t) = e^{ix_0(t)-\gamma(t)}, \!}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0(t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t)} are real functions with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0(t)} a homogeneous function of degree one and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(t)} a positive homogeneous function of degree one.^[25] More formally:^[25]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0(at) = ax_0(t),}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (at) = |a|\gamma (t),}

for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .

An example of a bivariate Cauchy distribution can be given by:^[26]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x, y; x_0,y_0,\gamma)= { 1 \over 2 \pi } \left[ { \gamma \over ((x - x_0)^2 + (y - y_0)^2 +\gamma^2)^{3/2} } \right] .}

Note that in this example, even though the covariance between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is 0, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are not statistically independent.^[26]

We also can write this formula for complex variable. Then the probability density function of complex cauchy is :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z; z_0,\gamma)= { 1 \over 2 \pi } \left[ { \gamma \over (|z-z_0|^2 +\gamma^2)^{3/2} } \right] .}

Analogous to the univariate density, the multidimensional Cauchy density also relates to the multivariate Student distribution. They are equivalent when the degrees of freedom parameter is equal to one. The density of a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} dimension Student distribution with one degree of freedom becomes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f({\mathbf x}; {\mathbf\mu},{\mathbf\Sigma}, k)= \frac{\Gamma\left(\frac{1+k}{2}\right)}{\Gamma(\frac{1}{2})\pi^{\frac{k}{2}}\left|{\mathbf\Sigma}\right|^{\frac{1}{2}}\left[1+({\mathbf x}-{\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x}-{\mathbf\mu})\right]^{\frac{1+k}{2}}} .}

Properties and details for this density can be obtained by taking it as a particular case of the multivariate Student density.

Transformation properties[edit | edit source]

• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(x_0,\gamma)} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle kX + \ell \sim \textrm{Cauchy}(x_0 k+\ell, \gamma |k|)} ^[27]
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(x_0, \gamma_0)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y \sim \operatorname{Cauchy}(x_1,\gamma_1)} are independent, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X+Y \sim \operatorname{Cauchy}(x_0+x_1,\gamma_0 +\gamma_1)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X-Y \sim \operatorname{Cauchy}(x_0-x_1, \gamma_0+\gamma_1)}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(0,\gamma)} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{1}{X} \sim \operatorname{Cauchy}(0, \tfrac{1}{\gamma})}
• McCullagh's parametrization of the Cauchy distributions:^[28] Expressing a Cauchy distribution in terms of one complex parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = x_0+i\gamma} , define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(\psi)} to mean Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(x_0,|\gamma|)} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(\psi)} then: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{aX+b}{cX+d} \sim \operatorname{Cauchy}\left(\frac{a\psi+b}{c\psi+d}\right)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} are real numbers.
• Using the same convention as above, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(\psi)} then:^[28] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{X-i}{X+i} \sim \operatorname{CCauchy}\left(\frac{\psi-i}{\psi+i}\right)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{CCauchy}} is the circular Cauchy distribution.

Lévy measure[edit | edit source]

The Cauchy distribution is the stable distribution of index 1. The Lévy–Khintchine representation of such a stable distribution of parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma } is given, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Stable}(\gamma, 0, 0)\,} by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{E}\left( e^{ixX} \right) = \exp\left( \int_{ \mathbb{R} } (e^{ixy} - 1) \Pi_\gamma(dy) \right)}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_\gamma(dy) = \left( c_{1, \gamma} \frac{1}{y^{1 + \gamma}} 1_{ \left\{y > 0\right\} } + c_{2,\gamma} \frac{1}{|y|^{1 + \gamma}} 1_{\left\{ y < 0 \right\}} \right) \, dy }

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1, \gamma}, c_{2, \gamma} } can be expressed explicitly.^[29] In the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 1 } of the Cauchy distribution, one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{1, \gamma} = c_{2, \gamma} } .

This last representation is a consequence of the formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi |x| = \operatorname{PV }\int_{\mathbb{R} \setminus\lbrace 0 \rbrace} (1 - e^{ixy}) \, \frac{dy}{y^2} }

Related distributions[edit | edit source]

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Cauchy}(0,1) \sim \textrm{t}(\mathrm{df}=1)\,} Student's t distribution
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Cauchy}(\mu,\sigma) \sim \textrm{t}_{(\mathrm{df}=1)}(\mu,\sigma)\,} non-standardized Student's t distribution
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X, Y \sim \textrm{N}(0,1)\, X, Y} independent, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac X Y\sim \textrm{Cauchy}(0,1)\,}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \textrm{U}(0,1)\,} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \left( \pi \left(X-\tfrac{1}{2}\right) \right) \sim \textrm{Cauchy}(0,1)\,}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Log-Cauchy}(0, 1)} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(X) \sim \textrm{Cauchy}(0, 1)}
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(x_0,\gamma)} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac1X \sim \operatorname{Cauchy}\left(\tfrac{x_0}{x_0^2+\gamma^2},\tfrac{\gamma}{x_0^2+\gamma^2}\right)}
• The Cauchy distribution is a limiting case of a Pearson distribution of type 4^{[citation needed]}
• The Cauchy distribution is a special case of a Pearson distribution of type 7.^[1]
• The Cauchy distribution is a stable distribution: if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \textrm{Stable}(1, 0, \gamma, \mu)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \operatorname{Cauchy}(\mu, \gamma)} .
• The Cauchy distribution is a singular limit of a hyperbolic distribution^{[citation needed]}
• The wrapped Cauchy distribution, taking values on a circle, is derived from the Cauchy distribution by wrapping it around the circle.
• If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \textrm{N}(0,1)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z \sim \operatorname{Inverse-Gamma}(1/2, s^2/2)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = \mu + X \sqrt Z \sim \operatorname{Cauchy}(\mu,s)} . For half-Cauchy distributions, the relation holds by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \sim \textrm{N}(0,1) I\{X>=0\}} .

Relativistic Breit–Wigner distribution[edit | edit source]

In nuclear and particle physics, the energy profile of a resonance is described by the relativistic Breit–Wigner distribution, while the Cauchy distribution is the (non-relativistic) Breit–Wigner distribution.^{[citation needed]}

Occurrence and applications[edit | edit source]

• In spectroscopy, the Cauchy distribution describes the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape. Many mechanisms cause homogeneous broadening, most notably collision broadening.^[30] Lifetime or natural broadening also gives rise to a line shape described by the Cauchy distribution.
• Applications of the Cauchy distribution or its transformation can be found in fields working with exponential growth. A 1958 paper by White ^[31] derived the test statistic for estimators of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\beta}} for the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{t+1}=\beta{x}_t+\varepsilon_{t+1},\beta>1} and where the maximum likelihood estimator is found using ordinary least squares showed the sampling distribution of the statistic is the Cauchy distribution.

• The Cauchy distribution is often the distribution of observations for objects that are spinning. The classic reference for this is called the Gull's lighthouse problem^[32] and as in the above section as the Breit–Wigner distribution in particle physics.
• In hydrology the Cauchy distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Cauchy distribution to ranked monthly maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
• The expression for imaginary part of complex electrical permittivity according to Lorentz model is a model VAR (value at risk) producing a much larger probability of extreme risk than Gaussian Distribution.^[33]

See also[edit | edit source]

• Lévy flight and Lévy process
• Laplace distribution, the Fourier transform of the Cauchy distribution
• Cauchy process
• Stable process
• Slash distribution

References[edit | edit source]

External links[edit | edit source]

• "Cauchy distribution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Earliest Uses: The entry on Cauchy distribution has some historical information.
• Weisstein, Eric W. "Cauchy Distribution". MathWorld.
• GNU Scientific Library – Reference Manual
• Ratios of Normal Variables by George Marsaglia