# Cauchy

##### The Cauchy Distribution

Density, distribution function, quantile function and random
generation for the Cauchy distribution with location parameter
`location`

and scale parameter `scale`

.

- Keywords
- distribution

##### Usage

```
dcauchy(x, location = 0, scale = 1, log = FALSE)
pcauchy(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qcauchy(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rcauchy(n, location = 0, scale = 1)
```

##### Arguments

- x, q
- vector of quantiles.
- p
- vector of probabilities.
- n
- number of observations. If
`length(n) > 1`

, the length is taken to be the number required. - location, scale
- location and scale parameters.
- log, log.p
- logical; if TRUE, probabilities p are given as log(p).
- lower.tail
- logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

##### Details

If `location`

or `scale`

are not specified, they assume
the default values of `0`

and `1`

respectively.

The Cauchy distribution with location $l$ and scale $s$ has density $$f(x) = \frac{1}{\pi s} \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}$$ for all $x$.

##### Value

`dcauchy`

,`pcauchy`

, and`qcauchy`

are respectively the density, distribution function and quantile function of the Cauchy distribution.`rcauchy`

generates random deviates from the Cauchy. The length of the result is determined by`n`

for`rcauchy`

, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than`n`

are recycled to the length of the result. Only the first elements of the logical arguments are used.

##### source

`dcauchy`

, `pcauchy`

and `qcauchy`

are all calculated
from numerically stable versions of the definitions.

`rcauchy`

uses inversion.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
*Continuous Univariate Distributions*, volume 1, chapter 16.
Wiley, New York.

##### See Also

Distributions for other standard distributions, including
`dt`

for the t distribution which generalizes
`dcauchy(*, l = 0, s = 1)`

.

##### Examples

`library(stats)`

`dcauchy(-1:4)`

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*