# Lognormal {stats}

The Log Normal Distribution
Package:
stats
Version:
R 3.0.2

### 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 ` ` 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 ≤ x], otherwise, P[X > x].

### Details

The log normal distribution has density f(x) = 1/(√(2 π) σ x) e^-((log x - μ)^2 / (2 σ^2)) where μ and σ are the mean and standard deviation of the logarithm. The mean is E(X) = exp(μ + 1/2 σ^2), the median is med(X) = exp(μ), and the variance Var(X) = exp(2*μ + σ^2)*(exp(σ^2) - 1) and hence the coefficient of variation is sqrt(exp(σ^2) - 1) which is approximately σ when that is small (e.g., σ < 1/2).

### Values

`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 parameters for the other functions. The numerical parameters other than `n` are recycled to the length of the result. Only the first elements of the logical parameters are used.

### 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.

### Note

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

Distributions for other standard distributions, including `dnorm` for the normal distribution.
`dlnorm(1) == dnorm(0)`