# Statistics - Gumbel Distribution

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Gumbel Distribution represents the distribution of extreme values either maximum or minimum of samples used in various distributions. It is used to model distribution of peak levels. For example, to show the distribution of peak temperatures of the year if there is a list of maximum temperatures of 10 years.

## Probability density function

Probability density function of Gumbel distribution is given as:

## Formula

${ P(x) = \frac{1}{\beta} e^{[\frac{x - \alpha}{\beta} - e^{\frac{x - \alpha}{\beta}}]} }$

Where −

• ${\alpha}$ = location parameter.

• ${\beta}$ = scale parameter.

• ${x}$ = random variable.

## Cumulative distribution function

Cumulative distribution function of Gumbel distribution is given as:

## Formula

${ D(x) = 1 - e^{-e^{\frac{x - \alpha}{\beta}}}}$

Where −

• ${\alpha}$ = location parameter.

• ${\beta}$ = scale parameter.

• ${x}$ = random variable.

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