This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

GumbelDistribution

 GumbelDistribution represents a Gumbel distribution with location parameter and scale parameter .
• The Gumbel distribution gives the asymptotic distribution of the minimum value in a sample from a distribution such as the normal distribution.
• The asymptotic distribution of the maximum value, also sometimes called a Gumbel distribution, is implemented in Mathematica as ExtremeValueDistribution.
• The probability density for value in a Gumbel distribution is proportional to . »
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (6)
Generate a set of pseudorandom numbers that have a Gumbel distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare the density histogram of the sample with the PDF of the estimated distribution:
Skewness and kurtosis are constant:
Different moments with closed forms as functions of parameters:
Closed form for symbolic order:
Hazard function:
Quantile function:
 Applications   (2)
The lifetime of a device has a Gumbel distribution. Find the reliability of the device:
The hazard function is exponentially increasing in time:
Find the reliability of two such devices in series:
Find the reliability of two such devices in parallel:
Compare the reliability of both systems for and :
The magnitude of the annual maximum earthquake can be modeled using GumbelDistribution. Consider earthquakes in the United States in the past 200 years:
Find the annual maximum:
Create a sample eliminating the missing data:
Fit a Gumbel distribution to the data:
Compare the histogram of the sample with the PDF of the estimated distribution:
Find the probability of the annual maximum earthquake having a magnitude of at least 6:
Find the average magnitude of the annual maximum earthquake:
Simulate the magnitudes of the maximum earthquake for 30 years:
Parameter influence on the CDF for each :
Gumbel distribution is closed under translation and scaling by a positive factor:
Skewness is the negative of the skewness of ExtremeValueDistribution:
ExtremeValueDistribution is skewed to the right, while GumbelDistribution is skewed to the left:
Kurtosis is the same as for ExtremeValueDistribution:
The family of Gumbel distribution is closed under a minimum:
The CDF of GumbelDistribution solves the minimum stability postulate equation:
Find and :
Relationships to other distributions:
Gumbel distribution is a transformation of ExtremeValueDistribution:
GumbelDistribution is a transformation of WeibullDistribution:
A truncated Gumbel distribution is a GompertzMakehamDistribution:
Gumbel distribution is a special case of MinStableDistribution:
Gumbel distribution is a transformation of MaxStableDistribution:
The difference of two variates from GumbelDistribution follows the same distribution as the difference of two variates from ExtremeValueDistribution, which is LogisticDistribution:
The sum of a Gumbel distribution and ExtremeValueDistribution follows LogisticDistribution:
The distribution of minimum values is given by GumbelDistribution:
The distribution of maximum values is given by ExtremeValueDistribution:
GumbelDistribution is not defined when is not a real number:
GumbelDistribution is not defined when is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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