Extreme value distributions

Extreme value distributions

Type-I - Gumbel distribution

The probability function is:

$$f(x)=\dfrac{1}{\beta }e^{-(\dfrac{x-\mu}{\beta }+e^{-\dfrac{x-\mu }{\beta }})}$$

We can use:

$$z=\dfrac{x-\mu }{\beta }$$

To get:

$$f(x)=\dfrac{1}{\beta }e^{-(z+e^{-z})}$$

The difference between two draws from a Gumbel distribution is drawn from the logistic function.