Gumbel_distribution: The Gumbel Distribution

In leem: Laboratory of Teaching to Statistics and Mathematics

The Gumbel Distribution

Description

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)
# CDF
pgumbel(1, 0, 1)
# Quantile
qgumbel(0.2, 0, 1)

leem documentation built on April 3, 2025, 6:04 p.m.