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Cauchy distribution

Profile

The Cauchy distribution is symmetric but has tails so heavy that its mean and variance do not exist. It is a classic warning against summarizing every symmetric cloud with a normal distribution.

Property Consequence
symmetry centre at x0
tails ~1/x² mean and variance undefined
ratio of normals natural probabilistic construction

Origin and terminology

The density appears in Augustin-Louis Cauchy’s work on errors and averages, and is also known as the Lorentz distribution in physics and the Breit–Wigner shape for resonances.

In real models

  • spectral lines and resonances in physics
  • ratio-error models and robustness demonstrations involving extreme tails

Comparisons

Do not use the mean, standard deviation, or moment-based diagnostics as if they were reliable summaries of a Cauchy sample.

The average that refuses to settle

Take independent Cauchy variables and compute their average: the result has exactly the same distribution. Increasing the sample size does not narrow the curve. This does not contradict the law of large numbers because one of its essential ingredients is missing: a Cauchy expectation does not exist.

The standard distribution also appears as the ratio of two independent standard Normal variables and as the Lorentz profile in spectroscopy. These constructions explain both its tails and its uses. In real data, an erratic sample mean can signal heavy tails but does not prove a Cauchy law. Medians and quantiles are more stable summaries for exploring it.

Decision guide

A good candidate when: a symmetric law with very frequent extremes is needed and location, rather than the mean, is the central quantity of interest.

Compare it with: Student’s t with estimated degrees of freedom. Cauchy fixes the extreme case ν=1; sample mean and variance do not stabilize as they do under a Normal model.

References

  • SciPy reference: scipy.stats.cauchy — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Cauchy, A. L. (1853). Sur les résultats moyens d’observations de même nature et sur les résultats les plus probables. Comptes Rendus de l’Académie des Sciences, 37, 611–618.

Cauchy Distribution: equations and calculator

Distribution defintion

X∼Cauchy(x0,γ)X\sim\mathrm{Cauchy}\left(x0,\gamma\right)

Distribution domain

x∈(−∞,+∞)x\in (-\infty,+\infty)

Parameters domain and parameters constraints

x0∈R,γ∈R+x_0\in\mathbb{R},\gamma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1πarctan⁡(x−x0γ)+12F_{X}\left(x\right)=\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}

Probability density function

fX(x)=1πγ [1+(x−x0γ)2]f_{X}\left(x\right)=\frac{1}{\pi\gamma\,\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}

Percent point function/Sample

FX−1(u)=x0+γ tan⁡[π(u−12)]F^{-1}_{X}\left(u\right)=x_0+\gamma\,\tan\left[\pi\left(u-\tfrac{1}{2}\right)\right]

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=undefined\mathrm{Mean}(X)=\text{undefined}

Parametric variance

Variance(X)=undefined\mathrm{Variance}(X)=\text{undefined}

Parametric skewness

Skewness(X)=undefined\mathrm{Skewness}(X)=\text{undefined}

Parametric kurtosis

Kurtosis(X)=undefined\mathrm{Kurtosis}(X)=\text{undefined}

Parametric median

Median(X)=x0\mathrm{Median}(X)=x_0

Parametric mode

Mode(X)=x0\mathrm{Mode}(X)=x_0

Additional information and definitions

x0:Location parameterx_0:\text{Location parameter}
γ:Scale parameter\gamma:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}