Gumbel distribution
Density, distribution function, quantile function and random generation for the Gumbel distribution.
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)
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 |
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))
Bury, K. (1999). Statistical Distributions in Engineering. Cambridge University Press.
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)
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