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Gumbel distribution

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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. For example we would use it to find the maximum level of a river in a particular year if we had the list of maximum values for the past ten years. It is therefore useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.

The distribution of the samples could be of the normal or exponential type. 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

p = e − e μ − x β . {\displaystyle p=e^{-e^{\frac {\mu -x}{\beta }}}.}

The Gumbel distribution is the case where μ = 0 and β = 1.

A more practical way of using the distribution could be

p = e − e − 0.367 A − x A − M ; {\displaystyle p=e^{-e^{-0.367{\frac {A-x}{A-M}}}};}
− 0.367 = ln ⁡ ( − ln ⁡ ( 0.5 ) ) {\displaystyle -0.367=\ln(-\ln(0.5))}

where M is the median. To fit values one could get the median straight away and then vary A until 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 ⁡ ( r n d ) ) {\displaystyle x=A-B\ln(-\ln(rnd))}

Its percentiles can be given by ;

x = A − B ∗ ln ⁡ ( − ln ⁡ ( p ) ) {\displaystyle x=A-B*\ln(-\ln(p))}

ie Q 1 = A − B ∗ ln ⁡ ( − ln ⁡ ( .25 ) ) {\displaystyle Q1=A-B*\ln(-\ln(.25))}

The median is A − B ∗ ln ⁡ ( − ln ⁡ ( 0.5 ) ) {\displaystyle A-B*\ln(-\ln(0.5))}

Q 3 = A − B ∗ ln ⁡ ( − ln ⁡ ( 0.75 ) ) {\displaystyle Q3=A-B*\ln(-\ln(0.75))}

The mean is A + g B {\displaystyle A+gB} where g {\displaystyle g} = Euler's constant = 0.57721.

The standard deviation is

B π 1 6 {\displaystyle B\pi {\sqrt {\frac {1}{6}}}}

Its mode is A.

See also

Category: