In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. Its importance arises from the fact that it is the limit distribution of the maxima of a sequence of independent and identically distributed random variables. Because of this, the GEV is used as an approximation to model the maxima of long (finite) sequences of random variables.

f(x,mu,sigma,k) = 1/sigma(1+kz)^(-1/k-1)exp(-(1+kz)^(-1/k))

where   

z=(x-mu)/sigma

support   

mu-sigma/k < x ⇐ inf, k>0   

-inf ⇐ x ⇐ mu-sigma/k, k

-inf ⇐ x ⇐ inf, k=0