Gumbel distribution

Take the maximum of a large number of independent draws from a “nice” distribution — Exponential, Normal, anything with a light upper tail. Shift and rescale. The limiting distribution is Gumbel.

f (x; μ, β) = \frac{1}{β} exp (- \frac{x - μ}{β} - e^{- (x - μ) / β})

Gumbel (mode 0.0, mean 0.58) Normal, same mean and variance

μ (location / mode) 0.0 β (scale) 1.0 n (samples) 1000

PDF

f (x; μ, β) = \frac{1}{β} exp (- \frac{x - μ}{β} - e^{- (x - μ) / β})

. Mean

E [X] = μ + β γ

sits right of the mode; right skew is the whole point — it's the distribution of the maximum of many draws.

What to notice

Right-skewed. The mode sits at $μ$ but the mean leans right by $β γ$ , where $γ$ is the Euler-Mascheroni constant ≈ 0.5772.
Matched-variance Normal is wrong on both sides. The dashed curve has the same mean and variance as the Gumbel but misses badly — it’s too light in the right tail and too heavy in the left.
Closed-form CDF. Unusually clean for an extreme-value distribution:

F (x) = exp (- e^{- (x - μ) / β})

Inverting gives the sampler this demo uses.

Extreme-value theory

Fisher and Tippett proved that the renormalised maximum of $n$ iid draws — when a non-degenerate limit exists — must be one of three types: Gumbel (light-tailed parents), Fréchet (heavy-tailed parents), or reverse-Weibull (bounded parents). These three collapse into the generalised extreme value (GEV) family.

Applications follow the theory. Insurers model maximum annual flood heights as Gumbel. Civil engineers size dams and sea-walls from Gumbel-fitted return periods. Meteorologists use it to talk about 100-year storms.

E [X] = μ + β γ (γ \approx 0.5772)

Var (X) = π^{2} β^{2} /6