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 ≤ 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).

### 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)