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# ExtremeValueDistribution

 ExtremeValueDistribution represents an extreme value distribution with location parameter and scale parameter .
• The extreme value distribution gives the asymptotic distribution of the maximum value in a sample from a distribution such as the normal distribution.
• The probability density for value in an extreme value distribution is proportional to . »
• The asymptotic distribution of the minimum value, also sometimes called an extreme value distribution, is implemented in Mathematica as GumbelDistribution. »
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 the extreme value distribution:
Compare the histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a 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   (3)
The lifetime of a device has an extreme value distribution. Find the reliability of the device:
The failure rate has a horizontal asymptote that depends only on the second parameter:
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 :
ExtremeValueDistribution can be used to model monthly maximum wind speeds:
Fit distribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the probability of monthly maximum wind exceeding 90 km/h:
Find average monthly maximum wind speed:
Simulate maximum wind speed for 30 months:
Construct an approximation for the distribution of maximum value in a normal sample of size :
Compare density function plots:
Mean of the approximation:
Compare with the mean of exact distribution:
Parameter influence on the CDF for each :
Extreme value distribution is closed under translation and scaling by a positive factor:
Skewness is the negative of the skewness of GumbelDistribution:
ExtremeValueDistribution is skewed to the right, while GumbelDistribution is skewed to the left:
Kurtosis is the same as for GumbelDistribution:
CDF of ExtremeValueDistribution solves the stability postulate equation:
Find the conditions on and such that the above is an identity:
Relationships to other distributions:
Extreme value distribution is a transformation of GumbelDistribution:
WeibullDistribution is a transformation of extreme value distribution:
Extreme value distribution is a special case of MaxStableDistribution:
Extreme value distribution is a transformation of MinStableDistribution:
Extreme value distribution is a transformation of ExponentialDistribution:
The difference of two variates from extreme value distribution follows the same distribution as the difference of two variates from GumbelDistribution, which is LogisticDistribution:
Sum of extreme value distribution and GumbelDistribution follows LogisticDistribution:
The distribution of maximum values is given by ExtremeValueDistribution:
The distribution of minimum values is given by GumbelDistribution:
ExtremeValueDistribution is not defined when is not a real number:
ExtremeValueDistribution 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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