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# S09 - IE 336 Feb 25 2009 Name Test#1 1 Let f(x y = c(x y)2...

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IE 336 Feb. 25, 2009 Name: Test #1 1. Let f ( x, y ) = ( c ( x + y ) 2 - 2 x 2 , 0 y 2 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 ( Y ) and E ( XY ). 1

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IE 336 Feb. 25, 2009 Name: 2. Let X be a stationary Markov chain with states { 4 , 7 , 8 } . Let P = q 2 r 0 . 5 0 . 3 p 0 . 4 p 0 . 5 q 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 4, the second row/column corresponds to state 7, and the third row/column corresponds to state 8). Clearly p 74 = P ( X 1 = 4 | X 0 = 7) = 0 . 3, p 87 = P ( X 1 = 7 | X 0 = 8) = 0 . 5, etc. (a) Compute p , q , and r and determine the transition diagram of this Markov chain. (b) Find P ( X 1 = 7 | X 0 = 4), P ( X 5 = 4 | X 4 = 8 , X 2 = 7), and P ( X 8 = 4 | X 5 = 4 , x 3 = 7). (c) Determine the walk probability p 7847844847 . (d) Assume that at some point the process is in state 7. Compute the expected number of visits to state 4 in the following three steps.
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