Editing Cauchy distribution (section)

==Occurrence and applications==
===In general===
[[File:Cauchy distribution.png|thumb|upright=1.15|Fitted cumulative Cauchy distribution to maximum one-day rainfalls using [[CumFreq]], see also [[distribution fitting]]<ref name="cumfreq">{{cite web |title=CumFreq, free software for cumulative frequency analysis and probability distribution fitting |url=https://www.waterlog.info/cumfreq.htm |url-status=live |archive-url=https://web.archive.org/web/20180221100105/https://www.waterlog.info/cumfreq.htm|archive-date=2018-02-21}}</ref>]]
*In [[spectroscopy]], the Cauchy distribution describes the shape of [[spectral line]]s 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 [[Line broadening#Pressure broadening|collision broadening]].<ref>{{cite book |author=E. Hecht |year=1987 |title=Optics |page=603 |edition=2nd |publisher=[[Addison-Wesley]] }}</ref>  [[Spectral line#Natural broadening|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 <ref>{{cite journal |author=White, J.S. |date=December 1958 |title=The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case |journal=The Annals of Mathematical Statistics |volume=29 |issue=4 |pages=1188–1197 |doi=10.1214/aoms/1177706450 |doi-access=free}}</ref> derived the test statistic for estimators of <math>\hat{\beta}</math> for the equation <math>x_{t+1}=\beta{x}_t+\varepsilon_{t+1},\beta>1</math> 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<ref>Gull, S.F. (1988) Bayesian Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin. https://doi.org/10.1007/978-94-009-3049-0_4 {{Webarchive|url=https://web.archive.org/web/20220125125834/https://link.springer.com/chapter/10.1007%2F978-94-009-3049-0_4 |date=2022-01-25 }}</ref> 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 position]]s as part of the [[cumulative frequency analysis]].
*The expression for the imaginary part of complex [[Permittivity|electrical permittivity]], according to the Lorentz model, is a Cauchy distribution.
*As an additional distribution to model [[fat tails]] in [[computational finance]], Cauchy distributions can be used to model VAR ([[value at risk]]) producing a much larger probability of extreme risk than [[Gaussian Distribution]].<ref>Tong Liu (2012), An intermediate distribution between Gaussian and Cauchy distributions. https://arxiv.org/pdf/1208.5109.pdf {{Webarchive|url=https://web.archive.org/web/20200624234315/https://arxiv.org/pdf/1208.5109.pdf |date=2020-06-24 }}</ref>

===Relativistic Breit–Wigner distribution===
{{Main article|Relativistic Breit–Wigner distribution}}
In [[nuclear physics|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|date=March 2011}}