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Cauchy

A heavy-tailed ratio of normals; its mean and variance do not exist.
Type continuousSupport ℝFamily Heavy tail
Probability densityf(x) = 1 / [πγ(1+((x−x₀)/γ)²)]
continuous

Cauchy in motion

-8.0-4.00.04.08.0
Supportℝ
Meanundefined
Varianceundefined
Connections

Where the cauchy
comes from — and leads.

Every link is one of the four moves from the relationship map: special cases, transformations, limits, and Bayesian conjugacy.

Uniform← tan π(U−½)

The Cauchy quantile transforms a standard uniform variate.

TransformationNormal← Z₁/Z₂

The ratio of two independent standard normals is standard Cauchy.

TransformationStudent t← ν = 1

Student t with one degree of freedom is standard Cauchy.

Special case

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