How to show that, for two random variables $X,Y\sim\text{Gumbel}[0,1]$, $X-Y\sim\text{Logistic}[0,1]$?
I tried to use the convolution formula $$\int_{-\infty}^{\infty}f_X(w)f_{-Y}(z-w)dw$$ I obtain $$\int_{-\infty}^{\infty} e^{-e^{-w}-e^{z-w}+z-2w}~dw$$ However, I cannot find the antiderivative of this expression.
Now I have two questions:
- Am I taking the right approach to show this?
- Can you give me a hint how to solve the integral or recommend any literature that might help?
The change of variables $u=\mathrm e^{-w}$ transforms this into a neat gamma integral, to which one can apply the change of variable $v=(\mathrm e^z-1)u$ to conclude.
Note that the transformation $w\to\mathrm e^{-w}$ is ubiquitous in quite a few Gumbel related manipulations.