ExtremeValueDistribution

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:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Closed form for symbolic order:

Hazard function:

Quantile function:

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:

ExtremeValueDistribution is a transformation of WeibullDistribution:

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: