Cauchy distribution

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.

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	${\displaystyle {\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!}$
CDF	${\displaystyle {\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!}$
Quantile	${\displaystyle x_{0}+\gamma \,\tan[\pi (p-{\tfrac {1}{2}})]}$
Mean	undefined
Median	${\displaystyle x_{0}\!}$
Mode	${\displaystyle x_{0}\!}$
Variance	undefined
MAD	${\displaystyle \gamma }$
Skewness	undefined
Ex. 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 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. 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.