# Lognormal

##### 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`

.

- Keywords
- distribution

##### 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)`

.

##### 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.

##### See Also

Distributions for other standard distributions, including
`dnorm`

for the normal distribution.

##### Examples

`library(stats)`

```
# NOT RUN {
dlnorm(1) == dnorm(0)
# }
```

*Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0*