The Log Normal Distribution

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

qlnorm2(p, mean = 1, sd = sqrt(exp(1) - 1), lower.tail = TRUE, log.p = FALSE)

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.

Arguments

p: vector of probabilities.
mean: the mean value on the true scale (vectorised)
sd: the standard deviation on the true scale (vectorised)
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
log.p: logical; if TRUE, probabilities p are given as log(p).

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., $\sigma < 1/2$).

Note

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

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.

Examples

qlnorm2(c(0.25,0.5,0.72), 2, 1)
#> [1] 1.300774 1.788854 2.355843