# 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.

##### 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)`

```
# NOT RUN {
dcauchy(-1:4)
# }
```

*Documentation reproduced from package stats, version 3.6.1, License: Part of R 3.6.1*