Cauchy

Probability density function The purple curve is the standard Cauchy distribution
Cumulative distribution function
Parameters	${\displaystyle x_{0}\!}$ location (real) ${\displaystyle \gamma >0}$ scale (real)
Support	${\displaystyle \displaystyle x\in (-\infty ,+\infty )\!}$
PDF	${\displaystyle {\frac {1}{\pi \gamma \,\left}}\!}$
CDF	${\displaystyle {\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!}$
Quantile	${\displaystyle x_{0}+\gamma \,\tan}$
Mean	undefined
Median	${\displaystyle x_{0}\!}$
Mode	${\displaystyle x_{0}\!}$
Variance	undefined
MAD	${\displaystyle \gamma }$
Skewness	undefined
Excess kurtosis	undefined
Entropy	${\displaystyle \log(4\pi \gamma )\!}$
MGF	does not exist
CF	${\displaystyle \displaystyle \exp(x_{0}\,i\,t-\gamma \,|t|)\!}$
Fisher information	${\displaystyle {\frac {1}{2\gamma ^{2}}}}$

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 ${\displaystyle (x_{0},\gamma )}$ 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 § 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 stable distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

History

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 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.

Constructions

Here are the most important constructions.

Rotational symmetry

If one stands in front of a line and kicks a ball with a direction (more precisely, an angle) uniformly at random towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution.

More formally, consider a point at ${\displaystyle (x_{0},\gamma )}$ in the x-y plane, and select a line passing the point, with its direction (angle with the ${\displaystyle x}$-axis) chosen uniformly (between -90° and +90°) at random. The intersection of the line with the x-axis is the Cauchy distribution with location ${\displaystyle x_{0}}$ and scale ${\displaystyle \gamma }$.

This definition gives a simple way to sample from the standard Cauchy distribution. Let ${\displaystyle u}$ be a sample from a uniform distribution from ${\displaystyle }$, then we can generate a sample, ${\displaystyle x}$ from the standard Cauchy distribution using

${\displaystyle x=\tan \left(\pi (u-{\frac {1}{2}})\right)}$

When ${\displaystyle U}$ and ${\displaystyle V}$Zdroj:https://en.wikipedia.org?pojem=Cauchy_distribution
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