# 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.`sdlog <= 0<="" code=""> is an error and returns ``NaN`

.

##### 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.

##### 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)`

```
dlnorm(1) == dnorm(0)
```

*Documentation reproduced from package stats, version 3.1.1, License: Part of R 3.1.1*