| % File src/library/stats/man/Lognormal.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{Lognormal} |
| \alias{Lognormal} |
| \alias{dlnorm} |
| \alias{plnorm} |
| \alias{qlnorm} |
| \alias{rlnorm} |
| \title{The Log Normal Distribution} |
| \description{ |
| Density, distribution function, quantile function and random |
| generation for the log normal distribution whose logarithm has mean |
| equal to \code{meanlog} and standard deviation equal to \code{sdlog}. |
| } |
| \usage{ |
| dlnorm(x, meanlog = 0, sdlog = 1, log = FALSE) |
| plnorm(q, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE) |
| qlnorm(p, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE) |
| rlnorm(n, meanlog = 0, sdlog = 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{meanlog, sdlog}{mean and standard deviation of the distribution |
| on the log scale with default values of \code{0} and \code{1} respectively.} |
| \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{dlnorm} gives the density, |
| \code{plnorm} gives the distribution function, |
| \code{qlnorm} gives the quantile function, and |
| \code{rlnorm} generates random deviates. |
| |
| The length of the result is determined by \code{n} for |
| \code{rlnorm}, 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. |
| } |
| \source{ |
| \code{dlnorm} is calculated from the definition (in \sQuote{Details}). |
| \code{[pqr]lnorm} are based on the relationship to the normal. |
| |
| Consequently, they model a single point mass at \code{exp(meanlog)} |
| for the boundary case \code{sdlog = 0}. |
| } |
| \details{ |
| The log normal distribution has density |
| \deqn{ |
| f(x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-(\log(x) - \mu)^2/2 \sigma^2}% |
| }{f(x) = 1/(\sqrt(2 \pi) \sigma x) e^-((log x - \mu)^2 / (2 \sigma^2))} |
| where \eqn{\mu} and \eqn{\sigma} are the mean and standard |
| deviation of the logarithm. |
| The mean is \eqn{E(X) = exp(\mu + 1/2 \sigma^2)}, |
| the median is \eqn{med(X) = exp(\mu)}, and the variance |
| \eqn{Var(X) = exp(2\mu + \sigma^2)(exp(\sigma^2) - 1)}{Var(X) = exp(2*\mu + \sigma^2)*(exp(\sigma^2) - 1)} |
| and hence the coefficient of variation is |
| \eqn{\sqrt{exp(\sigma^2) - 1}}{sqrt(exp(\sigma^2) - 1)} which is |
| approximately \eqn{\sigma} when that is small (e.g., \eqn{\sigma < 1/2}). |
| } |
| %% Mode = exp(max(0, mu - sigma^2)) |
| \note{ |
| The cumulative hazard \eqn{H(t) = - \log(1 - F(t))}{H(t) = - log(1 - F(t))} |
| is \code{-plnorm(t, r, lower = FALSE, log = TRUE)}. |
| } |
| \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 14. |
| Wiley, New York. |
| } |
| \seealso{ |
| \link{Distributions} for other standard distributions, including |
| \code{\link{dnorm}} for the normal distribution. |
| } |
| \examples{ |
| dlnorm(1) == dnorm(0) |
| } |
| \keyword{distribution} |