Cauchy: Create a Cauchy distribution

In distributions3: Probability Distributions as S3 Objects

Create a Cauchy distribution

Description

Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.

Usage

Cauchy(location = 0, scale = 1)

Arguments

location	The location parameter. Can be any real number. Defaults to 0.
scale	The scale parameter. Must be greater than zero (?). Defaults to 1.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let X be a Cauchy variable with mean ⁠location =⁠ x_0 and scale = \gamma.

Support: R, the set of all real numbers

Mean: Undefined.

Variance: Undefined.

Probability density function (p.d.f):

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

Cumulative distribution function (c.d.f):

F(t) = \frac{1}{\pi} \arctan \left( \frac{t - x_0}{\gamma} \right) + \frac{1}{2}

Moment generating function (m.g.f):

Does not exist.

Value

A Cauchy object.

See Also

Other continuous distributions: Beta(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.