FrechetDistribution
✖
FrechetDistribution
represents the Fréchet distribution with shape parameter α and scale parameter β.
represents the Fréchet distribution with shape parameter α, scale parameter β, and location parameter μ.
Details

- The Fréchet 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 Fréchet distribution is proportional to
for
and zero otherwise.
- The probability density for value
in a Fréchet distribution with location parameter is proportional to
for
and zero otherwise.
- FrechetDistribution allows α and β to be any positive real numbers and μ to be any real number.
- FrechetDistribution allows β and μ to be any quantities with the same unit dimensions and α to be a dimensionless quantity. »
- FrechetDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Background & Context
- FrechetDistribution[α,β,μ] represents a continuous statistical distribution defined over the interval
and parametrized by a real number μ (called a "location parameter") and two positive real numbers α and β (called a "shape parameter" and a "scale parameter", respectively). The probability density function (PDF) of a Fréchet distribution is unimodal, with the parameter μ determining the horizontal offset of the PDF and the parameters α and β together determining the PDF's overall height and steepness. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of
. (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Fréchet distribution is sometimes referred to as a type II extreme value distribution (not to be confused with "the" extreme value distribution, implemented in the Wolfram Language as ExtremeValueDistribution).
- FrechetDistribution is one of four distributions (along with GumbelDistribution, ExtremeValueDistribution, and WeibullDistribution) classified under the general heading "extreme value distributions" and is therefore used as a tool for quantifying "extreme" or "rare" events. Named for French mathematician Maurice René Fréchet, the Fréchet distribution was introduced in the late 1920s as a possible limit distribution for the largest order statistic (i.e. as a potential asymptotic distribution for the maximum value of a sample distributed according to some other distribution). Since its inception, the Fréchet distribution has been linked to the modeling of a number of real-world phenomena, including human lifetimes, radioactive emissions, flood and seismic analyses, and maximum one-day rainfalls.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Fréchet distribution. Distributed[x,FrechetDistribution[α,β,μ]], written more concisely as xFrechetDistribution[α,β,μ], can be used to assert that a random variable x is distributed according to a Fréchet distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability density and cumulative distribution functions may be given using PDF[FrechetDistribution[α,β,μ],x] and CDF[FrechetDistribution[α,β,μ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with a Fréchet distribution, EstimatedDistribution to estimate a Fréchet parametric distribution from given data, and FindDistributionParameters to fit data to a Fréchet distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Fréchet distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Fréchet distribution.
- TransformedDistribution can be used to represent a transformed Fréchet distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Fréchet distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Fréchet distributions.
- The Fréchet distribution is related to several other distributions. As mentioned previously, FrechetDistribution shares qualitative relationships with ExtremeValueDistribution, GumbelDistribution, and WeibullDistribution. These relationships can be quantified by noting that the PDF of FrechetDistribution[α,β] is precisely the same as TransformedDistribution[β2/z,zWeibullDistribution[α,β]], and that WeibullDistribution can be written as transformed versions of both ExtremeValueDistribution and GumbelDistribution. The PDF of FrechetDistribution[α,β] is the same as MaxStableDistribution[β,β/α,1/α] and MinStableDistribution[-β,β/α,1/α] under suitable transformations and assumptions and can be obtained from UniformDistribution under a suitable chain of transformations. FrechetDistribution is also related to ExpGammaDistribution, ExponentialDistribution, and LogisticDistribution.
Examples
open allclose allBasic Examples (5)Summary of the most common use cases

https://wolfram.com/xid/0pnbht9zlx22hoka-8nv957


https://wolfram.com/xid/0pnbht9zlx22hoka-09yruh


https://wolfram.com/xid/0pnbht9zlx22hoka-dg16n


https://wolfram.com/xid/0pnbht9zlx22hoka-guvsip


https://wolfram.com/xid/0pnbht9zlx22hoka-o45fre

Cumulative distribution function:

https://wolfram.com/xid/0pnbht9zlx22hoka-ocpmrk


https://wolfram.com/xid/0pnbht9zlx22hoka-swhxzv


https://wolfram.com/xid/0pnbht9zlx22hoka-r8zsko


https://wolfram.com/xid/0pnbht9zlx22hoka-haj0n0


https://wolfram.com/xid/0pnbht9zlx22hoka-4ttbqv


https://wolfram.com/xid/0pnbht9zlx22hoka-8mh43


https://wolfram.com/xid/0pnbht9zlx22hoka-7gjvmq


https://wolfram.com/xid/0pnbht9zlx22hoka-v0wyon


https://wolfram.com/xid/0pnbht9zlx22hoka-tpwe0s


https://wolfram.com/xid/0pnbht9zlx22hoka-6erttv


https://wolfram.com/xid/0pnbht9zlx22hoka-i923rg

Scope (8)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Fréchet distribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-qhtk5j
Compare its histogram to the PDF:

https://wolfram.com/xid/0pnbht9zlx22hoka-03mwaz

Distribution parameters estimation:

https://wolfram.com/xid/0pnbht9zlx22hoka-45b7g2
Estimate the distribution parameters from sample data:

https://wolfram.com/xid/0pnbht9zlx22hoka-epi747

Compare the density histogram of the sample with the PDF of the estimated distribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-f8ui5o

Skewness depends only on the first parameter:

https://wolfram.com/xid/0pnbht9zlx22hoka-rg8e7i


https://wolfram.com/xid/0pnbht9zlx22hoka-s7jglt


https://wolfram.com/xid/0pnbht9zlx22hoka-828scd


https://wolfram.com/xid/0pnbht9zlx22hoka-qp30pw


https://wolfram.com/xid/0pnbht9zlx22hoka-r6eqjl

Kurtosis depends only on the first parameter:

https://wolfram.com/xid/0pnbht9zlx22hoka-zzxgo9


https://wolfram.com/xid/0pnbht9zlx22hoka-bz3hru


https://wolfram.com/xid/0pnbht9zlx22hoka-00e8xe


https://wolfram.com/xid/0pnbht9zlx22hoka-hhqnw4


https://wolfram.com/xid/0pnbht9zlx22hoka-4tuwg0

Different moments with closed forms as functions of parameters:

https://wolfram.com/xid/0pnbht9zlx22hoka-js043h

https://wolfram.com/xid/0pnbht9zlx22hoka-rx074o

Closed form for symbolic order:

https://wolfram.com/xid/0pnbht9zlx22hoka-l5ytym


https://wolfram.com/xid/0pnbht9zlx22hoka-pknsqa


https://wolfram.com/xid/0pnbht9zlx22hoka-zg9ct4


https://wolfram.com/xid/0pnbht9zlx22hoka-9gzmth


https://wolfram.com/xid/0pnbht9zlx22hoka-e2svfd


https://wolfram.com/xid/0pnbht9zlx22hoka-1xgftb


https://wolfram.com/xid/0pnbht9zlx22hoka-7t24wm


https://wolfram.com/xid/0pnbht9zlx22hoka-t4upbb


https://wolfram.com/xid/0pnbht9zlx22hoka-wpdyuk


https://wolfram.com/xid/0pnbht9zlx22hoka-xnqfhj


https://wolfram.com/xid/0pnbht9zlx22hoka-146fsh


https://wolfram.com/xid/0pnbht9zlx22hoka-6aiaj4


https://wolfram.com/xid/0pnbht9zlx22hoka-dqi5us


https://wolfram.com/xid/0pnbht9zlx22hoka-yaor46

Consistent use of Quantity in parameters yields QuantityDistribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-fsj0zx


https://wolfram.com/xid/0pnbht9zlx22hoka-b4vv7f


https://wolfram.com/xid/0pnbht9zlx22hoka-ehm63

Applications (2)Sample problems that can be solved with this function
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:

https://wolfram.com/xid/0pnbht9zlx22hoka-viqmcx


https://wolfram.com/xid/0pnbht9zlx22hoka-8qrd2z

Find median annual maximal tephra volume:

https://wolfram.com/xid/0pnbht9zlx22hoka-niwg3t

Find the average annual maximal tephra volume:

https://wolfram.com/xid/0pnbht9zlx22hoka-1i762o

Find the mode of the distribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-86q2en

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

https://wolfram.com/xid/0pnbht9zlx22hoka-ghf70w

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

https://wolfram.com/xid/0pnbht9zlx22hoka-ejs6yd

FrechetDistribution can be used to model annual maximum wind speeds:

https://wolfram.com/xid/0pnbht9zlx22hoka-pcuqwr
Fit the distribution to the data:

https://wolfram.com/xid/0pnbht9zlx22hoka-hanor9

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

https://wolfram.com/xid/0pnbht9zlx22hoka-fp21xa

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

https://wolfram.com/xid/0pnbht9zlx22hoka-h4i4da

Find average annual maximum wind speed:

https://wolfram.com/xid/0pnbht9zlx22hoka-ga35ko

Simulate maximum wind speed for 30 years:

https://wolfram.com/xid/0pnbht9zlx22hoka-lpe5mq

Properties & Relations (8)Properties of the function, and connections to other functions
Fréchet distribution is closed under positive parameter scaling and translation:

https://wolfram.com/xid/0pnbht9zlx22hoka-trasct


https://wolfram.com/xid/0pnbht9zlx22hoka-78ic4z

The family of FrechetDistribution is closed under maximum:

https://wolfram.com/xid/0pnbht9zlx22hoka-2cnsmi


https://wolfram.com/xid/0pnbht9zlx22hoka-byqsg1

CDF of FrechetDistribution solves the maximum stability postulate equation:

https://wolfram.com/xid/0pnbht9zlx22hoka-lydrjv


https://wolfram.com/xid/0pnbht9zlx22hoka-nu5644

Relationships to other distributions:


https://wolfram.com/xid/0pnbht9zlx22hoka-lactu2


https://wolfram.com/xid/0pnbht9zlx22hoka-8jg4be


https://wolfram.com/xid/0pnbht9zlx22hoka-5rwbo

Fréchet distribution is a transformation of WeibullDistribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-c0xn66

Fréchet distribution is related to MaxStableDistribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-gmn7q0


https://wolfram.com/xid/0pnbht9zlx22hoka-xgf8bo


https://wolfram.com/xid/0pnbht9zlx22hoka-iydq66

Fréchet distribution is related to MinStableDistribution:

https://wolfram.com/xid/0pnbht9zlx22hoka-2itwt


https://wolfram.com/xid/0pnbht9zlx22hoka-39oopn


https://wolfram.com/xid/0pnbht9zlx22hoka-c92d5u

Wolfram Research (2010), FrechetDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FrechetDistribution.html (updated 2016).
Text
Wolfram Research (2010), FrechetDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FrechetDistribution.html (updated 2016).
Wolfram Research (2010), FrechetDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FrechetDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "FrechetDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/FrechetDistribution.html.
Wolfram Language. 2010. "FrechetDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/FrechetDistribution.html.
APA
Wolfram Language. (2010). FrechetDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrechetDistribution.html
Wolfram Language. (2010). FrechetDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrechetDistribution.html
BibTeX
@misc{reference.wolfram_2025_frechetdistribution, author="Wolfram Research", title="{FrechetDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/FrechetDistribution.html}", note=[Accessed: 09-May-2025
]}
BibLaTeX
@online{reference.wolfram_2025_frechetdistribution, organization={Wolfram Research}, title={FrechetDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/FrechetDistribution.html}, note=[Accessed: 09-May-2025
]}