Question: 2. (34 pts) Four people, A, B, C, D, form a line in random order. Let X be the number of people between A and

2. (34 pts) Four people, A, B, C, D, form a line in random order. Let X be the number of people between A and B, and Y be the number of people between C and D. a) b) C) d) ) (16 pts) Write down the joint pdf of X and Y, and find the marginal pdfs (you may write on the margins). (2 pts) From the joint pdf, find the probability that C and D are next to each other. (3 pts) From the joint pdf, find the probability that C and D are next to each other given that A and B are next to each other. (6 pts) Calculate the covariance of X and . Explain why its sign makes sense. (7 pts) Suppose A, B, C, D are among n people who form a line in random order. Let X be the number of people between A and B, and Y be the number of people between C and D. (i) (3pts) Find P(X =0,Y =0). (ii) (4 pts) Find P(Y = 0|X = 0)

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