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.

Cauchy(location = 0, scale = 1)

## Arguments

location The location parameter. Can be any real number. Defaults to 0. 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.

Other continuous distributions: Beta, ChiSquare, Exponential, FisherF, Gamma, LogNormal, Logistic, Normal, StudentsT, Tukey, Uniform, Weibull

## Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X#> Cauchy distribution (location = 10, scale = 0.2)
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] 0.0009940971log_pdf(X, 2)#> [1] -6.913676
cdf(X, 2)#> [1] 0.00795609quantile(X, 0.7)#> [1] 10.14531
cdf(X, quantile(X, 0.7))#> [1] 0.7quantile(X, cdf(X, 7))#> [1] 7