The Gumbel Distribution

Density, distribution function, quantile function and random generation for the normal distribution with parameters: location and scale

Usage

dgumbel(x, location, scale)

pgumbel(q, location, scale, lower.tail = TRUE)

qgumbel(p, location = 0, scale = 1, lower.tail = TRUE)

Arguments

x, q

vector of quantiles.

location

numerical. It represents location parameter. See Details.

scale

numerical. It represents scale parameter. See Details.

lower.tail

logical; if TRUE (default), probabilities are \(P[X \leq x]\) otherwise, \(P[X > x]\).

p

vector of probabilities.

Details

The CDF of Gumbel distribution is: $$ F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}, \quad \mu \in \mathbf{R}, \beta > 0, $$ where \(\mu\) is location parameter (location) and \(\beta\) is scale parameter (scale). The PDF of Gumbel distribution is: $$ \frac{1}{\beta }e^{-(z+e^{-z})}, $$ where \(z={\frac {x-\mu }{\beta }}\). The quantile is: $$ \mu -\beta \ln(-\ln(p)), \quad 0 < p < 1. $$

Examples

# PDF
dgumbel(1, 0, 1)
#> [1] 0.2546464
# CDF
pgumbel(1, 0, 1)
#> [1] 0.6922006
# Quantile
qgumbel(0.2, 0, 1)
#> [1] -0.475885