Can Z and U be considered independent in this scenario?

[lolypop]

Hi ,
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 .

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

lolypop

 

[bpet]

Z is discrete and doesn't have a pdf - one way around is to consider the cdf instead, i.e. show that the joint cdf of Z and U is a product of the marginal cdfs.