3.8 Suppose X and Y are independent random variables with X E(, 1) and Y ...

3.8 Suppose X and Y are independent random variables with X ∼ E(λ, 1) and Y ∼

E(µ, 1). It is impossible to obtain direct observations of X and Y . Instead, we observe the random variables Z and W, where Z = min{X, Y } and W =

1 if Z = X 0 if Z = Y.

Find the joint distribution of Z and W and show that they are independent. (The X and Y variables are censored., a situation that often arises in medical experiments. Suppose that X measures survival time from some treatment, and the patient leaves the survey for some unrelated reason. We do not get a measurement on X, but only a lower bound.)