Editing Cauchy distribution (section)

{{short description|Probability distribution}}
{{redirect-distinguish|Lorentz distribution|Lorenz curve|Lorenz system}}
{{Probability distribution
 | name       =Cauchy
 | type       =density
 | box_width  =300px
 | pdf_image  =[[File:cauchy pdf.svg|300px|Probability density function for the Cauchy distribution]]<br><small>The purple curve is the standard Cauchy distribution</small>
 | cdf_image  =[[File:cauchy cdf.svg|300px|Cumulative distribution function for the Cauchy distribution]]
 | parameters =<math>x_0\!</math> [[location parameter|location]] ([[real number|real]])<br><math>\gamma > 0</math> [[scale parameter|scale]] (real)
 | support    =<math>\displaystyle x \in (-\infty, +\infty)\!</math>
 | pdf        =<math>\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!</math>
 | cdf        =<math>\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}\!</math>
 | quantile   = <math>x_0+\gamma\,\tan[\pi(p-\tfrac{1}{2})]</math>
<!-- invalid parameter | qdf        =<math>\gamma\,\pi\,\sec^2(\pi\,(p-\tfrac{1}{2}))\!</math> -->
 | mean       =[[indeterminate form|undefined]]
 | median     =<math>x_0\!</math>
 | mode       =<math>x_0\!</math>
 | variance   =[[indeterminate form|undefined]]
 | mad        =<math>\gamma</math>
 | skewness   =[[indeterminate form|undefined]]
 | kurtosis   =[[indeterminate form|undefined]]
 | entropy    =<math>\log(4\pi\gamma)\!</math>
 | mgf        =does not exist
 | char       =<math>\displaystyle \exp(x_0\,i\,t-\gamma\,|t|)\!</math>
 | fisher     = <math>\frac{1}{2\gamma^2}</math>
}}
The '''Cauchy distribution''', named after [[Augustin-Louis Cauchy]], is a [[continuous probability distribution]].  It is also known, especially among [[physicist]]s, as the '''Lorentz distribution''' (after [[Hendrik Lorentz]]), '''Cauchy–Lorentz distribution''', '''Lorentz(ian) function''', or '''Breit–Wigner distribution'''.  The Cauchy distribution <math>f(x; x_0,\gamma)</math> is the distribution of the {{mvar|x}}-intercept of a ray issuing from <math>(x_0,\gamma)</math> with a uniformly distributed angle.  It is also the distribution of the [[Ratio distribution|ratio]] of two independent [[Normal distribution|normally distributed]] random variables with mean zero.

The Cauchy distribution is often used in statistics as the canonical example of a "[[pathological (mathematics)|pathological]]" distribution since both its [[expected value]] and its [[variance]] are undefined (but see {{slink||Moments}} below). The Cauchy distribution does not have finite [[moment (mathematics)|moment]]s of order greater than or equal to one; only fractional absolute moments exist.<ref name=jkb1>{{cite book|author1=N. L. Johnson |author2=S. Kotz |author3=N. Balakrishnan |title=Continuous Univariate Distributions, Volume 1|publisher=Wiley|location=New York|year=1994}}, Chapter 16.</ref> 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 distribution]]s with a probability density function that can be expressed analytically, the others being the [[normal distribution]] and the [[Lévy distribution]].