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In probability theory and statistics the Gumbel distribution is used to find the minimum (or the maximum) of a number of samples of various distributions.

These distributions could be of the normal or exponential type. It is used for the extreme values of water levels, floods and wind velocities. The Gumbel distribution, and similar distributions, are used in extreme value theory.

In particular, the Gumbel distribution is a special case of the Fisher-Tippett distribution, also known as the log-Weibull distribution, defined as

${\displaystyle p=\exp(-\exp((A-x)/B).}$

The Gumbel distribution is the case where A=0 and B=1.

A more practical way of using the distribution could be

     p=exp(-exp(-0.367*(A-x)/(A-M))  ;-.367=ln(-ln(.5))

where M is the Median.To fit values one could get the Median straight away and then vary A untill it fits the list of values.

Its variates(ie to get a list of random values) can be given as ;

      x=A-B*ln(-ln(rnd))

Its percentiles can be given by ;

      x=A-B*ln(-ln(p))

ie Q1=A-B*ln(-ln(.25))

The Median is A-B*ln(-ln(.5))

 Q3=A-B*ln(-ln(.75))

The mean is A+g*B 'g=Eulers constant = .57721

The sd = B * Pi()* sqr(1/6)

Its mode is A

See also:

• order statistic