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GumbelDistribution

GumbelDistribution[Alpha, Beta]
represents a Gumbel distribution with location parameter Alpha and scale parameter Beta.
  • The Gumbel distribution gives the asymptotic distribution of the minimum value in a sample from a distribution such as the normal distribution.
  • The probability density for value x in a Gumbel distribution is proportional to ⅇ^(-ⅇ^((x-alpha)/beta)+(x-alpha)/beta). »
  • The asymptotic distribution of the maximum value, also sometimes called a Gumbel distribution, is implemented in Mathematica as ExtremeValueDistribution. »
EXAMPLESCLOSE ALL
Basic Examples   (2)
The mean and variance of a Gumbel distribution:
Probability density function:
The mean and variance of a Gumbel distribution:
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Probability density function:
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Scope   (4)
Generate a set of pseudorandom numbers that have the Gumbel distribution:
Properties based on higher-order moments:
Third moment of a Gumbel distribution:
The q^(th) quantile of a Gumbel distribution:
Applications   (2)
Plot the cumulative distribution function of a Gumbel distribution:
A contour plot as both x and Beta are varied:
Properties & Relations   (4)
The probability density function integrates to unity:
Moments can be obtained from the characteristic function:
The negative of a Gumbel random variable follows an ExtremeValueDistribution:
GumbelDistribution is exponentially related to WeibullDistribution:
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 Alpha is not a real number:
GumbelDistribution is not defined when Beta is not a positive real number:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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