1% File src/library/stats/man/Cauchy.Rd
2% Part of the R package, https://www.R-project.org
3% Copyright 1995-2014 R Core Team
4% Distributed under GPL 2 or later
5
6\name{Cauchy}
7\alias{Cauchy}
8\alias{dcauchy}
9\alias{pcauchy}
10\alias{qcauchy}
11\alias{rcauchy}
12\title{The Cauchy Distribution}
13\description{
14  Density, distribution function, quantile function and random
15  generation for the Cauchy distribution with location parameter
16  \code{location} and scale parameter \code{scale}.
17}
18\usage{
19dcauchy(x, location = 0, scale = 1, log = FALSE)
20pcauchy(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
21qcauchy(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
22rcauchy(n, location = 0, scale = 1)
23}
24\arguments{
25  \item{x, q}{vector of quantiles.}
26  \item{p}{vector of probabilities.}
27  \item{n}{number of observations. If \code{length(n) > 1}, the length
28    is taken to be the number required.}
29  \item{location, scale}{location and scale parameters.}
30  \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).}
31  \item{lower.tail}{logical; if TRUE (default), probabilities are
32    \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.}
33}
34\value{
35  \code{dcauchy}, \code{pcauchy}, and \code{qcauchy} are respectively
36  the density, distribution function and quantile function of the Cauchy
37  distribution.  \code{rcauchy} generates random deviates from the
38  Cauchy.
39
40  The length of the result is determined by \code{n} for
41  \code{rcauchy}, and is the maximum of the lengths of the
42  numerical arguments for the other functions.
43
44  The numerical arguments other than \code{n} are recycled to the
45  length of the result.  Only the first elements of the logical
46  arguments are used.
47}
48\details{
49  If \code{location} or \code{scale} are not specified, they assume
50  the default values of \code{0} and \code{1} respectively.
51
52  The Cauchy distribution with location \eqn{l} and scale \eqn{s} has
53  density
54  \deqn{f(x) = \frac{1}{\pi s}
55    \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}%
56  }{f(x) = 1 / (\pi s (1 + ((x-l)/s)^2))}
57  for all \eqn{x}.
58}
59\source{
60  \code{dcauchy}, \code{pcauchy} and \code{qcauchy} are all calculated
61  from numerically stable versions of the definitions.
62
63  \code{rcauchy} uses inversion.
64}
65\references{
66  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
67  \emph{The New S Language}.
68  Wadsworth & Brooks/Cole.
69
70  Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
71  \emph{Continuous Univariate Distributions}, volume 1, chapter 16.
72  Wiley, New York.
73}
74\seealso{
75  \link{Distributions} for other standard distributions, including
76  \code{\link{dt}} for the t distribution which generalizes
77  \code{dcauchy(*, l = 0, s = 1)}.
78}
79\examples{
80dcauchy(-1:4)
81}
82\keyword{distribution}
83