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 ???:uhh: lolypop
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.