# Gumbel: Gumbel distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

Density, distribution function, quantile function and random generation for the Gumbel distribution.

## Usage

 ```1 2 3 4 5 6 7``` ```dgumbel(x, mu = 0, sigma = 1, log = FALSE) pgumbel(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) qgumbel(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) rgumbel(n, mu = 0, sigma = 1) ```

## Arguments

 `x, q` vector of quantiles. `mu, sigma` location and scale parameters. Scale must be positive. `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]. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

## Details

Probability density function

f(x) = 1/σ * exp(-((x-μ)/σ + exp(-(x-μ)/σ)))

Cumulative distribution function

F(x) = exp(-exp(-(x-μ)/σ))

Quantile function

F^-1(p) = μ - σ * log(-log(p))

## References

Bury, K. (1999). Statistical Distributions in Engineering. Cambridge University Press.

## Examples

 ```1 2 3 4 5 6``` ```x <- rgumbel(1e5, 5, 2) hist(x, 100, freq = FALSE) curve(dgumbel(x, 5, 2), 0, 25, col = "red", add = TRUE) hist(pgumbel(x, 5, 2)) plot(ecdf(x)) curve(pgumbel(x, 5, 2), 0, 25, col = "red", lwd = 2, add = TRUE) ```

extraDistr documentation built on Sept. 7, 2020, 5:09 p.m.