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hw3 - IE 336 Feb 7 2011 Handout#4 Due Feb 14 2011 Homework...

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IE 336 Handout #4 Feb. 7, 2011 Due Feb. 14, 2011 Homework Set #3 1. As in problem 4 from homework 1, let f ( x, y ) = 2 e - x - y , 0 y x < be the joint pdf of two random variables X and Y . Find E ( Y ), E ( X ), and E ( Y | x ). 2. Let { X 0 , X 1 , X 2 , ... } be a stationary Markov chain. Further, let the set of real numbers S = { a, 2 , 3 } be the set of all possible states of this chain (clearly, a 6 = 2 and a 6 = 3). For any x S let p xx = 1 2 x . Further let p a 2 = p a 3 = p 32 = 3 7 and p 2 a = 0 . 2. (a) Find the real number a . (b) Determine P ( X 9 = a | X 8 = a, X 7 = 2 , X 6 = 3), P ( X 4 = a | X 3 = 3 , X 2 = 3 , X 1 = a ), and P ( X 4 = 2 | X 3 = 2 , X 1 = a ). (c) Determine the transition matrix and the transition diagram of this Markov chain. (d) If at epoch 7 the process is in state a how likely is that at epoch 8 the process will be in state a ? 3. Consider a truck problem similar to the one that we considered in class. Assume that instead of 4 cities we now have 3 cities, Chicago, Dallas, and Philadelphia. There is a truck driver who makes one trip between these cities every day. He starts the trip in one of them and finishes in
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