I am given X has distribution f(x)= (1/σ)*e^(-x/σ) and Y has the same distribution but with μ and y in place of sigma and x. I am also given Z=min{X,Y} and W={1 if Z=X and 0 if Z=Y}.

I am told to find the joint distribution of Z and W and also prove Z and W are independent.

I am stuck on this one. Does it break down into two cases (Z=X and Z=Y)?