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Cauchy Distribution

NIST/SEMATECH Section 1.3.6.6.3 Cauchy Distribution

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Cauchy(0, 1) PDF -10 -8 -6 -4 -2 0 2 4 6 8 10 x 0 0.05 0.1 0.15 0.2 0.25 0.3 f(x)

CDF

Cauchy(0, 1) CDF -10 -8 -6 -4 -2 0 2 4 6 8 10 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

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

Cumulative Distribution Function

F(x)=1πarctan⁡ ⁣(x−x0γ)+12F(x) = \frac{1}{\pi}\arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}

Properties

Mean

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Variance

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Overview

The Cauchy distribution is a symmetric distribution with heavy tails, centred at location x0x_0 with scale γ\gamma. Its mean and variance are undefined, making it a canonical example of a pathological distribution in statistics.

Related Distributions

Student's t-Distribution Normal Distribution