Extreme Value Distribution

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.
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The extreme value (Type I) distribution (the term first used by Lieblein, 1953) has the probability density function:

where
a is the location parameter
b is the scale parameter
e is the base of the natural logarithm, sometimes called Euler's e (2.71...)
This distribution is also sometimes referred to as the distribution of the largest extreme. See also, Process Analysis .
Extreme Value Distribution (animation)
The graphic above shows the shape of the extreme value distribution when the location parameter equals 0 and the scale parameter equals 1.