4. Customers arrive and enter a supermarket according to a Poisson process {N(t), t > 0} with rate l. For n e Z+, let Sn be the waiting time until the nth customer enters the supermarket. Upon arrival, the nth customer spends a length of time in the supermarket that is a continuous ry (denoted by Wn) with cdf F(w) = P(Wn 0. Assume that W1,W2, W3, ... are iid rvs which are also independent of the arrival process. 5] (a) Let X(t) be the number of customers still in the supermarket at time t, and let Y(t) be the number of customers who arrived and subsequently left the supermarket by time t. Determine the joint probability distribution of X(t) and Y(t). Are X(t) and Y(t) independent? 3] (b) Let m,n e Z+ such that 1 0} with rate l. For n e Z+, let Sn be the waiting time until the nth customer enters the supermarket. Upon arrival, the nth customer spends a length of time in the supermarket that is a continuous ry (denoted by Wn) with cdf F(w) = P(Wn 0. Assume that W1,W2, W3, ... are iid rvs which are also independent of the arrival process. 5] (a) Let X(t) be the number of customers still in the supermarket at time t, and let Y(t) be the number of customers who arrived and subsequently left the supermarket by time t. Determine the joint probability distribution of X(t) and Y(t). Are X(t) and Y(t) independent? 3] (b) Let m,n e Z+ such that 1