distributions3 (version 0.2.1)

Cauchy: 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)

Value

A Cauchy object.

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.

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))