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)

location | The location parameter. Can be any real number. Defaults
to |
---|---|

scale | The scale parameter. Must be greater than zero (?). Defaults
to |

A `Cauchy`

object.

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`

#> Cauchy distribution (location = 10, scale = 0.2)#> [1] 9.982203 10.053876 9.916324 10.336325 10.167877 10.626557 10.046357 #> [8] 10.001540 10.091892 10.137681#> [1] 0.0009940971#> [1] -6.913676#> [1] 0.00795609#> [1] 10.14531#> [1] 0.7#> [1] 7