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.