Cauchy {distributions3} | R Documentation |

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 |

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.

A `Cauchy`

object.

Other continuous distributions:
`Beta()`

,
`ChiSquare()`

,
`Erlang()`

,
`Exponential()`

,
`FisherF()`

,
`Frechet()`

,
`GEV()`

,
`GP()`

,
`Gamma()`

,
`Gumbel()`

,
`LogNormal()`

,
`Logistic()`

,
`Normal()`

,
`RevWeibull()`

,
`StudentsT()`

,
`Tukey()`

,
`Uniform()`

,
`Weibull()`

```
set.seed(27)
X <- Cauchy(10, 0.2)
X
mean(X)
variance(X)
skewness(X)
kurtosis(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))
```

[Package *distributions3* version 0.2.1 Index]