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# F09 - IE 336 Oct 7 2009 Name Test#1 1 Let cyex f(x y =...

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IE 336 Oct. 7, 2009 Name: Test #1 1. Let f ( x, y ) = cye - x - 2 x 0 , 0 y 1 cye - 2 x 0 x 1 , 0 y 1 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 ( XY ). 1

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IE 336 Oct. 7, 2009 Name: 2. Let X be a stationary Markov chain with states { 2 , 5 , 9 } . Let P = 2 q 3 r p r q 4 r 2 + 3 q 4 2 p q 3 r 2 be the one-step transition matrix of the Markov chain X (rows and columns of matrix P are ordered in the same way the states are, i.e. the first row/column corresponds to state 2, the second row/column corresponds to state 5, and the third row/column corresponds to state 9). Clearly p 52 = P ( X 1 = 2 | X 0 = 5) = r , p 29 = P ( X 1 = 9 | X 0 = 2) = p , etc. (a) If q + r = 5 6 compute p , q , and r and determine the transition diagram of this Markov chain. (b) Find P ( X 5 = 9 | X 4 = 5), P ( X 8 = 2 | X 7 = 2 , X 2 = 5), and P ( X 8 = 5 | X 4 = 5 , x 5 = 9).
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