| % File src/library/stats/man/Exponential.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Exponential} |
| \alias{Exponential} |
| \alias{dexp} |
| \alias{pexp} |
| \alias{qexp} |
| \alias{rexp} |
| \title{The Exponential Distribution} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the exponential distribution with rate \code{rate} |
| (i.e., mean \code{1/rate}). |
| } |
| \usage{ |
| dexp(x, rate = 1, log = FALSE) |
| pexp(q, rate = 1, lower.tail = TRUE, log.p = FALSE) |
| qexp(p, rate = 1, lower.tail = TRUE, log.p = FALSE) |
| rexp(n, rate = 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{rate}{vector of rates.} |
| \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{dexp} gives the density, |
| \code{pexp} gives the distribution function, |
| \code{qexp} gives the quantile function, and |
| \code{rexp} generates random deviates. |
| |
| The length of the result is determined by \code{n} for |
| \code{rexp}, 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{rate} is not specified, it assumes the default value of |
| \code{1}. |
| |
| The exponential distribution with rate \eqn{\lambda} has density |
| \deqn{f(x) = \lambda {e}^{- \lambda x}} for \eqn{x \ge 0}. |
| } |
| \source{ |
| \code{dexp}, \code{pexp} and \code{qexp} are all calculated |
| from numerically stable versions of the definitions. |
| |
| \code{rexp} uses |
| |
| Ahrens, J. H. and Dieter, U. (1972). |
| Computer methods for sampling from the exponential and normal distributions. |
| \emph{Communications of the ACM}, \bold{15}, 873--882. |
| } |
| \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 19. |
| Wiley, New York. |
| } |
| \seealso{ |
| \code{\link{exp}} for the exponential function. |
| |
| \link{Distributions} for other standard distributions, including |
| \code{\link{dgamma}} for the gamma distribution and |
| \code{\link{dweibull}} for the Weibull distribution, both of which |
| generalize the exponential. |
| } |
| \note{ |
| The cumulative hazard \eqn{H(t) = - \log(1 - F(t))}{H(t) = - log(1 - F(t))} |
| is \code{-pexp(t, r, lower = FALSE, log = TRUE)}. |
| } |
| \examples{ |
| dexp(1) - exp(-1) #-> 0 |
| |
| ## a fast way to generate *sorted* U[0,1] random numbers: |
| rsunif <- function(n) { n1 <- n+1 |
| cE <- cumsum(rexp(n1)); cE[seq_len(n)]/cE[n1] } |
| plot(rsunif(1000), ylim=0:1, pch=".") |
| abline(0,1/(1000+1), col=adjustcolor(1, 0.5)) |
| } |
| \keyword{distribution} |