1. (Uniform Order Statistics). Suppose X1;::: ; X5 are independent U Œ0; 1] variables. Find the joint density of X.2/; X.3/; X.4/, and E.X.4/ C X.2/ - 2X.3//.
2. (Exponential Order Statistics). Suppose X; Y; Z are three inde- pendent standard exponential variables, and let U; V; W be their minimum, median, and maximum. Find the densities of U; V; W; W - U .
3. * (Waiting Time). Peter, Paul, and Mary went to a bank to do some business. Two counters were open, and Peter and Paul went first. Peter, Paul, and Mary will each take, independently, an Exp.?/amount of time to finish their business from the moment they arrive at the counter.
(a) What is the density of the epoch of the last departure?
(b) What is the probability that Mary will be the last to finish?
(c) What is the density of the total time taken by Mary from arrival to finishing her business?