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I got this question in my midterm today but up till now I don't know how to solve it ,

The Question is as follow :

If X and Y are two exponential Rv with different lambda . and there's a new Rvs Z and U are defined such that :

Z= 0 : X<Y and 1 : X=> Y

U= min(X,Y)

and the Question asked to proof that Z and U are independent .

So I started my solution by deriving the pdf of U since I know how to then tried to derive the pdf of Z but didn't know where to start and got stuck , since I didn't how to get the pdf function when the boundary of the Rv is other RV.

Can anyone tell me of a way to derive the pdf of Z . or is there another way to solve the problem ???

lolypop