Probability Playground: The Gumbel Distribution

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, ..., X_n 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 f_X(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) = \frac{1}{β} e^{- (z + e^{- z})}, z = \frac{x - μ}{β}

F (x; μ, β) = e^{- e^{- (x - μ) / β}}

S (x; μ, β) = 1 - e^{- e^{- (x - μ) / β}}

h (x; μ, β) = \frac{1}{β} \frac{e^{- (z + e^{- z})}}{1 - e^{- e^{- z}}}, z = \frac{x - μ}{β}

Support

- ∞ < x < ∞

Mean

Variance

Example	μ	β
Let X₁, ..., X_n 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)

μ = β =

E(X) = , Var(X) =

Note that the pdf of the Gumbel distribution is right-skewed, with a steep left side and a long right tail.

Since the pdf of the Gumbel distribution is right-skewed, the cdf rise steeply at first then more slowly.

The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).

Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.

The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.

Note that h(x) converges to 1/β as x → ∞.

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 X_i + 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 reduced by βlog n 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 X_i + μ − βlog n, where max X_i 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)

E(Y) = , Var(Y) =

Proof

X ∼ Gumbel(μ, β)

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

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