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The Gumbel distribution is used to model the distribution of the maximum of a sample. It is therefore a type of extreme value distribution (type I), along with the Fréchet and Weibull.

The distribution arises naturally from a sample X₁, ..., Xn drawn from an exponential(β) distribution. The distribution of max{Xᵢ} + μ - βlog n converges to a Gumbel(μ, β) distribution as n → ∞. In fact, the suitably normalized maximum of any sample {Xᵢ} will converge to a Gumbel random variable if the pdf fX(x) falls faster than a power law as x → ∞.

Parameter Range Description
μ -∞ < μ < ∞ Location parameter
β β > 0 Scale parameter

Probability Density Function

f ( x | μ , β ) = 1 β e − ( z + e − z ) , z = x − μ β

Support

− ∞ < x < ∞

Mean

Variance

Example μ β
Let X₁, ..., Xn be drawn from an exponential(1) distribution. Then the distribution of max{Xᵢ} - log n converges to a Gumbel(0, 1) distribution as n → ∞. 0.000 1.000
Maximum daily snowfall (in feet) in Buffalo, NY for the years 1893 - 2021 follows an approximate Gumbel(0.68, 0.26) distribution. 0.6800 0.2600
Annual peak streamflow (in 1000 ft3/sec) for the Little Tonawanda Creek at Linden, NY for the years 1913 - 2005 follows an approximate Gumbel(0.77, 0.51) distribution. 0.7700 0.5100

X ~ Gumbel(μ, β)

Chart of the Gumbel distribution Chart area for displaying the Gumbel pdf, cdf, visualization, and simulation

E(X) = , Var(X) =

Note that this distribution is right-skewed, with a steep left side and a long right tail.

The illustration above shows a sample of n points chosen independently and at random from an exponential(β) distribution. The distribution of the adjusted maximum max Xi + μ - βlog n converges to a Gumbel(μ, β) distribution as n approaches infinity, where μ denotes a location parameter. Note that the maximum needs to be scaled since it otherwise increases without bound as n increases.

The simulation above shows a sample of n points (marked red) chosen independently and at random from an exponential(β) distribution. The light blue line shows the value of max Xi + μ - βlog n, where max Xi is the maximum value of the sample and μ denotes a location parameter. The distribution of this adjusted maximum converges to a Gumbel(μ, β) distribution as n approaches infinity. The histogram accumulates the results of each simulation.

Y = max Xᵢ ~ Gumbel(μ + βlog n, β) X1 - X2 ~ Logistic(μ1 - μ2, β) e-X ~ Weibull(1/β, e-μ)

Chart of the related distribution Chart area for displaying the related pdf, cdf, visualization, and simulation

E(Y) = , Var(Y) =

Note that taking the maximum just shifts the location of the distribution without changing the scale.

The logistic distribution therefore arises naturally as the limit of the difference of the maxima of two iid exponential distributions.

Note that:

  • The exponential transformation maps the location of the Gumbel to the scale of the Weibull and the scale of the Gumbel to the shape of the Weibull.
  • e-X is a decreasing function. So, as the probability mass of X shifts to the right, the probability mass of Y shifts to the left (and vice-versa).