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The Gumbel distribution is a type of probability distribution used to model extreme values.

The Gumbel distribution is also known as the Type I extreme value distribution. It is used to model the distribution of the maximum or minimum of a large number of independent and identically distributed random variables. The distribution is often used in hydrology, meteorology, and engineering to model extreme events such as floods, storms, and earthquakes.

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

f(x) = (1/β) exp(-(x-μ+exp(-(x-μ))/β))

where μ is the location parameter and β is the scale parameter. The cumulative distribution function (CDF) of the Gumbel distribution is given by:

F(x) = exp(-exp(-(x-μ)/β))

The Gumbel distribution has a number of useful properties. For example, the maximum of n independent and identically distributed random variables with a continuous distribution approaches a Gumbel distribution as n approaches infinity. This is known as the extreme value theorem.

In summary, the Gumbel distribution is a useful tool for modelling extreme events. It is widely used in a range of fields, including hydrology, meteorology, and engineering. The distribution is characterised by its location and scale parameters, and has a number of useful properties that make it a valuable tool for statistical analysis.

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