Cauchy distribution

E239287

continuous probability distribution heavy-tailed distribution probability distribution stable distribution univariate distribution

The Cauchy distribution is a continuous probability distribution with heavy tails and undefined mean and variance, often used as a classic example of pathological behavior in probability theory and statistics.

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All labels observed (3)

Label	Occurrences
Cauchy distribution canonical	3
Breit–Wigner distribution	1
Lorentz distribution	1

Statements (53)

Predicate	Object
instanceOf	continuous probability distribution ⓘ heavy-tailed distribution ⓘ probability distribution ⓘ stable distribution ⓘ univariate distribution ⓘ
belongsToFamily	location–scale family ⓘ
correspondsToStudentTWithDegreesOfFreedom	1 ⓘ
definedOn	real line ⓘ
hasAllMoments	do not exist ⓘ
hasAlternativeName	Cauchy distribution ⓘ surface form: Breit–Wigner distribution Cauchy distribution ⓘ surface form: Lorentz distribution
hasCharacteristicFunction	φ(t) = exp(i x0 t - γ \|t\|) ⓘ
hasConvolutionProperty	sum of independent Cauchy variables is Cauchy ⓘ
hasCumulativeDistributionFunction	F(x) = 1/π arctan(x) + 1/2 ⓘ F(x; x0, γ) = 1/π arctan((x - x0)/γ) + 1/2 ⓘ
hasEntropy	H = ln(4πγ) ⓘ
hasHazardFunction	h(x) = f(x) / (1 - F(x)) for x in ℝ ⓘ
hasKurtosis	undefined ⓘ
hasLocationParameter	0 ⓘ x0 ⓘ
hasMean	undefined ⓘ
hasMedian	x0 ⓘ
hasMedianAbsoluteDeviation	γ ⓘ
hasMode	x0 ⓘ
hasMomentGeneratingFunction	does not exist ⓘ
hasProbabilityDensityFunction	f(x) = 1 / [π (1 + x^2)] ⓘ f(x; x0, γ) = 1 / [πγ (1 + ((x - x0)/γ)^2)] ⓘ
hasQuantileFunction	Q(p) = x0 + γ tan[π(p - 1/2)] ⓘ
hasScaleParameter	1 ⓘ γ ⓘ
hasSkewness	0 ⓘ
hasStabilityIndex	1 ⓘ
hasStandardForm	standard Cauchy distribution ⓘ
hasSupport	(-∞, ∞) ⓘ
hasTailBehavior	f(x) ~ 1/(πγ) · 1/x^2 as \|x\| → ∞ ⓘ
hasVariance	undefined ⓘ
isHeavyTailed	true ⓘ
isLevyAlphaStable	true ⓘ
isSpecialCaseOf	Lévy alpha-stable distribution ⓘ Pearson distribution ⓘ surface form: Pearson type VII distribution Student’s t-distribution ⓘ surface form: Student's t-distribution generalized hyperbolic distribution ⓘ q-Gaussian distribution ⓘ
isStable	true ⓘ
isSymmetricAbout	x0 ⓘ
isUsedAs	example of distribution with undefined mean ⓘ example of distribution with undefined variance ⓘ example of pathological behavior in probability theory ⓘ heavy-tailed prior distribution ⓘ
isUsedIn	Bayesian inference ⓘ surface form: Bayesian statistics robust statistics ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
requiresScaleParameterCondition	γ > 0 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy → knownFor → Cauchy distribution ⓘ

Augustin-Louis → notableFor → Cauchy distribution ⓘ

subject surface form: Augustin-Louis Cauchy

Cauchy distribution → hasAlternativeName → Cauchy distribution ⓘ

this entity surface form: Lorentz distribution

Cauchy distribution → hasAlternativeName → Cauchy distribution ⓘ

this entity surface form: Breit–Wigner distribution

Tukey's lambda distribution → approximates → Cauchy distribution ⓘ