Lognormal {stats} R Documentation

## 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].

### 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).

### 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.

### Note

The cumulative hazard H(t) = - \log(1 - F(t)) is -plnorm(t, r, lower = FALSE, log = TRUE).

### Source

dlnorm is calculated from the definition (in ‘Details’). [pqr]lnorm are based on the relationship to the normal.

Consequently, they model a single point mass at exp(meanlog) for the boundary case sdlog = 0.

### 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.
dlnorm(1) == dnorm(0)