BUILT-IN MATHEMATICA SYMBOL

GumbelDistribution

represents a Gumbel distribution with location parameter

and scale parameter

.

MORE INFORMATION

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 . »

GumbelDistribution allows to be any real number and to be any positive real number.

GumbelDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

EXAMPLES

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Basic Examples (4)

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:

Properties & Relations (15)

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:

GumbelDistribution is a special case of ExpGammaDistribution:

Possible Issues (3)

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: