[What does the sentence above mean? Does it mean the Gumbel distribution is the probability distribution of the minimum or maximum of a sample? If so, it should say so. "... used to find the minimum ..." is not at all the same and does not make sense. "... of a number of samples ..."? What does that mean? Is the word "sample" being used to mean one observation within a sampele?]
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, whose cumulative distribution function is
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 ;
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