Gumbel Distribution
X ~ Gumbel(μ, β)
Extreme-Value Family • Published July 22, 2026
The light-tailed member of the Fisher-Tippett extreme-value family and the classic model for the maximum of many independent light-tailed observations: history from Gumbel's own flood/drought work, a max-of-Exponentials convergence result confirmed by simulation, a rare fully closed-form CDF, a clean exact tie to the already-published Exponential (e^-X for X~Gumbel(0,1)), no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
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About This Reference Sheet
The light-tailed member of the Fisher-Tippett extreme-value family and the classic model for the maximum of many independent light-tailed observations: history from Gumbel's own flood/drought work, a max-of-Exponentials convergence result confirmed by simulation, a rare fully closed-form CDF, a clean exact tie to the already-published Exponential (e^-X for X~Gumbel(0,1)), no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
Support
x ∈ (-∞, ∞)
Parameters
μ ∈ ℝ (location), β > 0 (scale)