distributions3 (version 0.1.1)

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)

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.

Value

A Cauchy object.

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, Exponential, FisherF, Gamma, LogNormal, Logistic, Normal, StudentsT, Tukey, Uniform, Weibull

Examples

# NOT RUN {
set.seed(27)

X <- Cauchy(10, 0.2)
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))
# }