When a function of two random variables is Gumbel? Consider two random variables $X,Y$. 
Is there any example in which $X$ and $Y$ have a known parametric distribution such that $f(X,Y)$ is Gumbel with scale $\sigma$ and location $\beta$, for some function $f$ (such as $X+Y$ for example)?
 A: Let $X,Y$ have some joint distribution, such that some $Z=t(X,Y)$ has a known (fully specified) continuous distribution. Let $F_Z$ be the cdf of $Z$. Let  $G_0$ be the cdf of a standard Gumbel.
Then $f_{0}(X,Y) = G_0^{-1}(F_Z(t(X,Y)))$ has a standard Gumbel distribution. Any other Gumbel may be obtained by considering $f_{\alpha,\tau}(X,Y) = \alpha+\tau G_0^{-1}(F_Z(t(X,Y)))$.
Thus we can construct any number of examples with ease.
For example, let $X$ and $Y$ be independent standard normals, let $\Phi$ be the standard normal cdf. Then $G_0^{-1}(\Phi(\frac{X+Y}{\sqrt{2}}))$ will be standard Gumbel.
Note that $G_0^{-1}(u) = -\log(-\log(u))$ is very simple in form.
Let us simulate, so you can see it in action:
 n <- 1000000L
 x <- rnorm(n)
 y <- rnorm(n)
 v <- -log(-log(pnorm((x+y)/sqrt(2))))
 hist(v,n=100,freq=FALSE)
 curve(exp(-(x+exp(-x))),-3,10,col=4,add=TRUE)


Another example would be to take $X,Y$ to be independent standard uniform, and let $Z=\max(X,Y)^2$, then $V=-\log(-\log(Z))$ will again be standard Gumbel.
 n=1000000L
 hist(-log(-log(pmax(runif(n),runif(n))^2)),n=100,freq=FALSE)
 curve(exp(-(x+exp(-x))),-3,10,col=4,add=TRUE)

(I won't include the plot this time, it looks the same as before).
... and so forth. You could construct a hundred examples in an afternoon, were you so inclined.
