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The Frechet distribution is a type of probability distribution used in extreme value theory.

The Frechet distribution is also known as the Type II extreme value distribution. It is used to model the distribution of the maximum of a large number of independent and identically distributed random variables. The distribution is characterized by its shape parameter, which determines the tail behavior of the distribution. The larger the shape parameter, the heavier the tail of the distribution.

The probability density function (PDF) of the Frechet distribution is given by:

f(x) = (α/σ) (x/σ)^{-α-1} e^{-(x/σ)^{-α}}

where x > 0, α > 0 is the shape parameter, and σ > 0 is the scale parameter. The cumulative distribution function (CDF) is given by:

F(x) = e^{-(x/σ)^{-α}}

The mean and variance of the Frechet distribution are given by:

μ = σΓ(1-1/α)

σ^2 = σ^2Γ(1-2/α) - μ^2

where Γ is the gamma function.

The Frechet distribution is commonly used in fields such as hydrology, finance, and engineering to model extreme events such as floods, stock market crashes, and earthquakes.

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