Create a Cauchy distribution

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(), 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
#> [1] "Cauchy(location = 10, scale = 0.2)"

mean(X)
#> [1] NaN
variance(X)
#> [1] NaN
skewness(X)
#> [1] NaN
kurtosis(X)
#> [1] NaN

random(X, 10)
#>  [1]  9.982203 10.053876  9.916324 10.336325 10.167877 10.626557 10.046357
#>  [8] 10.001540 10.091892 10.137681

pdf(X, 2)
#> [1] 0.0009940971
log_pdf(X, 2)
#> [1] -6.913676

cdf(X, 2)
#> [1] 0.00795609
quantile(X, 0.7)
#> [1] 10.14531

cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 7))
#> [1] 7