# GumbelDistribution

GumbelDistribution[α,β]

represents a Gumbel distribution with location parameter α and scale parameter β.

represents a Gumbel distribution with location parameter 0 and scale parameter 1.

# Details

• 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 the Wolfram Language as ExtremeValueDistribution. »
• The probability density for value in a Gumbel distribution is proportional to . »
• GumbelDistribution allows α to be any real number and β to be any positive real number.
• GumbelDistribution allows α and β to be any quantities of the same unit dimensions. »
• 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:

## Scope(7)

Generate a sample of pseudorandom numbers from 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:

Consistent use of Quantity in parameters yields QuantityDistribution:

Compute mean and standard deviation of the price:

## 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 annual maximum earthquake for 30 years:

## Properties & Relations(15)

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 GumbelDistribution is closed under a minimum:

The CDF of GumbelDistribution solves the minimum stability postulate equation:

Find and :

Relationships to other distributions:

GumbelDistribution is the negative of ExtremeValueDistribution:

GumbelDistribution is a transformation of WeibullDistribution:

A truncated Gumbel distribution is a GompertzMakehamDistribution:

GumbelDistribution is a special case of MinStableDistribution:

GumbelDistribution 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:

ParameterMixtureDistribution of ExtremeValueDistribution with Gumbel distribution follows LogisticDistribution:

## 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 α is not a real number:

GumbelDistribution is not defined when β is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

## Neat Examples(1)

PDFs for different α values with CDF contours:

Wolfram Research (2007), GumbelDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GumbelDistribution.html (updated 2016).

#### Text

Wolfram Research (2007), GumbelDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GumbelDistribution.html (updated 2016).

#### CMS

Wolfram Language. 2007. "GumbelDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/GumbelDistribution.html.

#### APA

Wolfram Language. (2007). GumbelDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GumbelDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_gumbeldistribution, author="Wolfram Research", title="{GumbelDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GumbelDistribution.html}", note=[Accessed: 13-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_gumbeldistribution, organization={Wolfram Research}, title={GumbelDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/GumbelDistribution.html}, note=[Accessed: 13-June-2024 ]}