How to show that the difference of two Gumbel distributed random variables makes a Logistic distribution?

[NAGG]

How to show that, for two random variables X∼Gumbel[a,b] and Y∼Gumbel[c,b], X−Y∼Logistic[a-c,b]? Can anyone show step by step how to make solution? Thank you!

 

[HallsofIvy]

You might want to start by defining "Gumbel Distribution" and "Logistic Distribution".

 

[NAGG]

You might want to start by defining "Gumbel Distribution" and "Logistic Distribution".

I know that for Logistic Distribution I have:


For Gumbel distribution I have:


I know that I need to get: F(x-y)=e^{x-y}/(1+e^{x-y}).