The Cauchy distribution

Source: R/dist_cauchy.R

[Stable]

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.

dist_cauchy(location, scale)

Arguments

location, scale

location and scale parameters.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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

stats::Cauchy

Examples

dist <- dist_cauchy(location = c(0, 0, 0, -2), scale = c(0.5, 1, 2, 1))

dist
#> <distribution[4]>
#> [1] Cauchy(0, 0.5) Cauchy(0, 1)   Cauchy(0, 2)   Cauchy(-2, 1) 
mean(dist)
#> [1] NA NA NA NA
variance(dist)
#> [1] NA NA NA NA
skewness(dist)
#> [1] NA NA NA NA
kurtosis(dist)
#> [1] NA NA NA NA

generate(dist, 10)
#> [[1]]
#>  [1]  0.05787226 -0.55679129  0.40127920  0.02515888  0.21376813 -0.75566519
#>  [7] -1.04594872  0.82954458  1.35507308 -0.65515529
#> 
#> [[2]]
#>  [1] -1.3979201 -0.2378976  7.7240494 -3.2162135  0.5704357 -4.6870245
#>  [7] -0.3316107 -3.2698299 -0.5842548 -3.3093403
#> 
#> [[3]]
#>  [1] -1.6654672  6.0118352 -1.0199788 -2.1051892  3.1036589  0.7319927
#>  [7]  0.6574569 -1.4521469  5.0489551  0.3341891
#> 
#> [[4]]
#>  [1]  -2.042410  -4.945683  -1.495282 -10.171663  -1.582879  -2.116187
#>  [7]  -1.851471  -1.444466  -2.202221  -2.092369
#> 

density(dist, 2)
#> [1] 0.03744822 0.06366198 0.07957747 0.01872411
density(dist, 2, log = TRUE)
#> [1] -3.284796 -2.754168 -2.531024 -3.977943

cdf(dist, 4)
#> [1] 0.9604166 0.9220209 0.8524164 0.9474315

quantile(dist, 0.7)
#> [1]  0.3632713  0.7265425  1.4530851 -1.2734575