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# F08 - IE 336 Oct 8 2008 Name Test#1 1 Let f(x y = y e3x 0 0...

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IE 336 Oct. 8, 2008 Name: Test #1 1. Let f ( x, y ) = ( ye - 3 x 0 x ≤ ∞ , 3 y c 0 otherwise be the joint pdf of two continuous random variables X and Y . (a) Determine the value of the constant c . (b) Are the random variables X and Y independent? Explain. (c) Compute E ( X | y ), E ( Y | x ), and E ( XY ). 1

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IE 336 Oct. 8, 2008 Name: 2. Let X be an ergodic Markov chain with states { 1 , 2 , 3 } . Let P = 0 . 1 0 . 5 0 . 4 0 . 3 0 . 1 0 . 6 0 . 1 0 . 1 0 . 8 be the one-step transition matrix of the Markov chain X . Clearly p 13 = P ( X 1 = 3 | X 0 = 1) = 0 . 4, p 32 = P ( X 1 = 2 | X 0 = 3) = 0 . 1, etc. (a) Determine P ( X 1 = 1 | X 0 = 2) and P ( X 6 = 2 | X 5 = 1 , X 4 = 3). (b) Compute P ( X 7 = 2 | X 5 = 3). (c) Find the walk probability p 1321112 . (d) Assume that at some point the process is in state 3. Compute the expected number of visits to state 1 in the following two steps. (e) Assume that at time step k the process is two times more likely to be in state 1 than in state 3. Also, assume that at the same time step k the process is two times less likely to be in state 2 than in state 1. Find the probability that at time step k +2 the process is in state 3.
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