# S10 - IE 336 Mar. 1, 2010 Name: Test #1 1. Let N be an even...

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IE 336 Mar. 1, 2010 Name: Test #1 1. Let N be an even positive integer. Further, let p ( x ) = cx 1 x N 3 2 cx N 3 + 1 x N 0 otherwise be the pdf of a discrete random variable X (the values x can take are integers). (a) Determine the value of the constant c . (b) Compute E ( X ). (Each of the answers should be function only of N .) 1

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IE 336 Mar. 1, 2010 Name: 2. Let X be a stationary Markov chain with states { 3 , 5 , 7 } . Let P = 2 q r p 3 q 4 r r 2 + 9 q 4 q 2 p 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 3, the second row/column corresponds to state 5, and the third row/column corresponds to state 7). Clearly p 55 = P ( X 1 = 5 | X 0 = 5) = r , p 37 = P ( X 1 = 7 | X 0 = 3) = p , etc. (a) If 3 q + r = 5 6 compute p , q , and r and determine the transition diagram of this Markov chain. (b) Find
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## S10 - IE 336 Mar. 1, 2010 Name: Test #1 1. Let N be an even...

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