BUILT-IN MATHEMATICA SYMBOL

FrechetDistribution

represents the Frechet distribution with shape parameter

and scale parameter

.

FrechetDistribution

represents the Frechet distribution with shape parameter

, scale parameter

, and location parameter

.

MORE INFORMATION

The Frechet distribution gives the asymptotic distribution of the maximum value in a sample from a distribution such as the Cauchy distribution.

FrechetDistribution is also known as type II extreme value distribution.

The probability density for value in a Frechet distribution is proportional to for and zero otherwise.

The probability density for value in a Frechet distribution with location parameter is proportional to for and zero otherwise.

FrechetDistribution allows and to be any positive real numbers and to be any real number.

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

EXAMPLES

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Basic Examples (5)

Probability density function:

With location parameter:

Cumulative distribution function:

With location parameter:

Mean:

Variance:

Median:

Probability density function:

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With location parameter:

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Cumulative distribution function:

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With location parameter:

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Mean:

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Variance:

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Median:

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Scope (7)

Generate a set of pseudorandom numbers that are Frechet distributed:

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 depends only on the first parameter:

Limiting values:

Kurtosis depends only on the first parameter:

Limiting values:

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

FactorialMoment:

Cumulant:

Hazard function:

With location parameter:

Quantile function:

With location parameter:

Applications (2)

According to a study, the annual maximal tephra (solid material) volume in volcanic eruptions follows a FrechetDistribution with shape parameter 0.71 and scale parameter 6.3, given in cubic kilometers:

Find median annual maximal tephra volume:

Find the average annual maximal tephra volume:

Find the mode of the distribution:

Find the probability that the annual maximal tephra volume is greater than 30 cubic kilometers:

Simulate the annual maximal tephra volume for the next 30 years:

FrechetDistribution can be used to model annual maximum wind speeds:

Fit the distribution into the data:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the probability of annual maximum wind exceeding 90 km/h:

Find average annual maximum wind speed:

Simulate maximum wind speed for 30 years:

Properties & Relations (9)

Parameter influence on the CDF for each

:

Frechet distribution is closed under positive parameter scaling and translation:

The family of FrechetDistribution is closed under maximum:

CDF of FrechetDistribution solves the maximum stability postulate equation:

Fix

and

and simplify:

Relationships to other distributions:

The default location is 0:

Frechet distribution is a transformation of WeibullDistribution:

Frechet distribution is related to MaxStableDistribution:

Frechet distribution is related to MinStableDistribution: