"Lorentz distribution" redirects here. Not to be confused with Lorenz curve or Lorenz system.
Cauchy
Probability density function
The purple curve is the standard Cauchy distribution
Cumulative distribution function
Parameters
location (real) scale (real)
Support
PDF
CDF
Quantile
Mean
undefined
Median
Mode
Variance
undefined
MAD
Skewness
undefined
Excess kurtosis
undefined
Entropy
MGF
does not exist
CF
Fisher information
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 is the distribution of the x-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 (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.
^N. L. Johnson; S. Kotz; N. Balakrishnan (1994). Continuous Univariate Distributions, Volume 1. New York: Wiley., Chapter 16.
and 21 Related for: Cauchy distribution information
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Global Information
![{\displaystyle {\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa7448ba911130c1e33621f1859393d3f00af5c)
![{\displaystyle x_{0}+\gamma \,\tan[\pi (p-{\tfrac {1}{2}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b28bd2a0c25cb1d212b29f0fc22baf6f84e3e0f)
