1% File src/library/stats/man/Lognormal.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{Lognormal}
7\alias{Lognormal}
8\alias{dlnorm}
9\alias{plnorm}
10\alias{qlnorm}
11\alias{rlnorm}
12\title{The Log Normal Distribution}
13\description{
14  Density, distribution function, quantile function and random
15  generation for the log normal distribution whose logarithm has mean
16  equal to \code{meanlog} and standard deviation equal to \code{sdlog}.
17}
18\usage{
19dlnorm(x, meanlog = 0, sdlog = 1, log = FALSE)
20plnorm(q, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE)
21qlnorm(p, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE)
22rlnorm(n, meanlog = 0, sdlog = 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{meanlog, sdlog}{mean and standard deviation of the distribution
30    on the log scale with default values of \code{0} and \code{1} respectively.}
31  \item{log, log.p}{logical; if TRUE, probabilities p are given as log(p).}
32  \item{lower.tail}{logical; if TRUE (default), probabilities are
33    \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.}
34}
35\value{
36  \code{dlnorm} gives the density,
37  \code{plnorm} gives the distribution function,
38  \code{qlnorm} gives the quantile function, and
39  \code{rlnorm} generates random deviates.
40
41  The length of the result is determined by \code{n} for
42  \code{rlnorm}, and is the maximum of the lengths of the
43  numerical arguments for the other functions.
44
45  The numerical arguments other than \code{n} are recycled to the
46  length of the result.  Only the first elements of the logical
47  arguments are used.
48}
49\source{
50  \code{dlnorm} is calculated from the definition (in \sQuote{Details}).
51  \code{[pqr]lnorm} are based on the relationship to the normal.
52
53  Consequently, they model a single point mass at \code{exp(meanlog)}
54  for the boundary case \code{sdlog = 0}.
55}
56\details{
57  The log normal distribution has density
58  \deqn{
59    f(x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-(\log(x) - \mu)^2/2 \sigma^2}%
60  }{f(x) = 1/(\sqrt(2 \pi) \sigma x) e^-((log x - \mu)^2 / (2 \sigma^2))}
61  where \eqn{\mu} and \eqn{\sigma} are the mean and standard
62  deviation of the logarithm.
63  The mean is \eqn{E(X) = exp(\mu + 1/2 \sigma^2)},
64  the median is \eqn{med(X) = exp(\mu)}, and the variance
65  \eqn{Var(X) = exp(2\mu + \sigma^2)(exp(\sigma^2) - 1)}{Var(X) = exp(2*\mu + \sigma^2)*(exp(\sigma^2) - 1)}
66  and hence the coefficient of variation is
67  \eqn{\sqrt{exp(\sigma^2) - 1}}{sqrt(exp(\sigma^2) - 1)} which is
68  approximately \eqn{\sigma} when that is small (e.g., \eqn{\sigma < 1/2}).
69}
70%% Mode = exp(max(0, mu - sigma^2))
71\note{
72  The cumulative hazard \eqn{H(t) = - \log(1 - F(t))}{H(t) = - log(1 - F(t))}
73  is \code{-plnorm(t, r, lower = FALSE, log = TRUE)}.
74}
75\references{
76  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
77  \emph{The New S Language}.
78  Wadsworth & Brooks/Cole.
79
80  Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
81  \emph{Continuous Univariate Distributions}, volume 1, chapter 14.
82  Wiley, New York.
83}
84\seealso{
85  \link{Distributions} for other standard distributions, including
86  \code{\link{dnorm}} for the normal distribution.
87}
88\examples{
89dlnorm(1) == dnorm(0)
90}
91\keyword{distribution}
92