stats (version 3.6.2)

# Lognormal: The Log Normal Distribution

## Description

Density, distribution function, quantile function and random generation for the log normal distribution whose logarithm has mean equal to meanlog and standard deviation equal to 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

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

meanlog, sdlog

mean and standard deviation of the distribution on the log scale with default values of 0 and 1 respectively.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $$P[X \le x]$$, otherwise, $$P[X > x]$$.

## Value

dlnorm gives the density, plnorm gives the distribution function, qlnorm gives the quantile function, and rlnorm generates random deviates.

The length of the result is determined by n for rlnorm, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

## Details

The log normal distribution has density $$f(x) = \frac{1}{\sqrt{2\pi}\sigma x} e^{-(\log(x) - \mu)^2/2 \sigma^2}%$$ where $$\mu$$ and $$\sigma$$ are the mean and standard deviation of the logarithm. The mean is $$E(X) = exp(\mu + 1/2 \sigma^2)$$, the median is $$med(X) = exp(\mu)$$, and the variance $$Var(X) = exp(2\mu + \sigma^2)(exp(\sigma^2) - 1)$$ and hence the coefficient of variation is $$\sqrt{exp(\sigma^2) - 1}$$ which is approximately $$\sigma$$ when that is small (e.g., $$\sigma < 1/2$$).

## References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 14. Wiley, New York.

Distributions for other standard distributions, including dnorm for the normal distribution.
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