1. A delivery truck travels from point A to point B and back using the same route eachday. There are four traffic

lights on the route. Let X denote the number of red lights thetruck encounters going from A to B and Y denote the number of red lights encounteredon the return trip. Data collected over a long period suggest that the joint PMF for Xand Y is given byy0 1 2 3 40 0.01 0.01 0.03 0.07 0.011 0.03 0.05 0.08 0.03 0.02x 2 0.03 0.11 0.15 0.01 0.013 0.02 0.07 0.10 0.03 0.014 0.01 0.06 0.03 0.01 0.01(a) Give the marginal PMFs of X and Y .(b) Find the expected value and the standard deviation of X.(c) Find the expected value and the standard deviation of Y .(d) Give the conditional distribution of X given Y = 3.(e) Find E[X|Y = 3].(f) If the truck encountered only one red light going from A to B on a day, what is theexpected number of red lights that the truck will encounter on the return trip?(g) Are X and Y independent? Give reason.(h) Find corr(X, Y ).

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