| % File src/library/stats/man/Cauchy.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Cauchy} |
| \alias{Cauchy} |
| \alias{dcauchy} |
| \alias{pcauchy} |
| \alias{qcauchy} |
| \alias{rcauchy} |
| \title{The Cauchy Distribution} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the Cauchy distribution with location parameter |
| \code{location} and scale parameter \code{scale}. |
| } |
| \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{ |
| \item{x, q}{vector of quantiles.} |
| \item{p}{vector of probabilities.} |
| \item{n}{number of observations. If \code{length(n) > 1}, the length |
| is taken to be the number required.} |
| \item{location, scale}{location and scale parameters.} |
| \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).} |
| \item{lower.tail}{logical; if TRUE (default), probabilities are |
| \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.} |
| } |
| \value{ |
| \code{dcauchy}, \code{pcauchy}, and \code{qcauchy} are respectively |
| the density, distribution function and quantile function of the Cauchy |
| distribution. \code{rcauchy} generates random deviates from the |
| Cauchy. |
| |
| The length of the result is determined by \code{n} for |
| \code{rcauchy}, and is the maximum of the lengths of the |
| numerical arguments for the other functions. |
| |
| The numerical arguments other than \code{n} are recycled to the |
| length of the result. Only the first elements of the logical |
| arguments are used. |
| } |
| \details{ |
| If \code{location} or \code{scale} are not specified, they assume |
| the default values of \code{0} and \code{1} respectively. |
| |
| The Cauchy distribution with location \eqn{l} and scale \eqn{s} has |
| density |
| \deqn{f(x) = \frac{1}{\pi s} |
| \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}% |
| }{f(x) = 1 / (\pi s (1 + ((x-l)/s)^2))} |
| for all \eqn{x}. |
| } |
| \source{ |
| \code{dcauchy}, \code{pcauchy} and \code{qcauchy} are all calculated |
| from numerically stable versions of the definitions. |
| |
| \code{rcauchy} uses inversion. |
| } |
| \references{ |
| Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) |
| \emph{The New S Language}. |
| Wadsworth & Brooks/Cole. |
| |
| Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) |
| \emph{Continuous Univariate Distributions}, volume 1, chapter 16. |
| Wiley, New York. |
| } |
| \seealso{ |
| \link{Distributions} for other standard distributions, including |
| \code{\link{dt}} for the t distribution which generalizes |
| \code{dcauchy(*, l = 0, s = 1)}. |
| } |
| \examples{ |
| dcauchy(-1:4) |
| } |
| \keyword{distribution} |