midterm2_Solution_Prob_Thy

midterm2_Solution_Prob_Thy - MATHEMATICS 2603 Midterm,...

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Unformatted text preview: MATHEMATICS 2603 Midterm, November 27, 2009 Show all your work. Use back of page if necessary. You can use your own BLANK scratch paper. Name : ID : 1. [20 points] State i of a Markov chain has period d . Does that mean P ( d ) ii > 0? Prove it or disprove it. Solution: It does not mean P ( d ) ii > 0. Consider any Markov chain with states { , 1 , 2 } . Assume that P 01 ,P 10 ,P 12 ,P 20 > 0 and all the other transition probabilities are zero. Then one checks that state 0 has period d = 1, however P 00 = 0. 1 2. [20 points] Prove that a random variable X is exponentially distributed with parameter μ if and only if, for any t ≥ 0, P { X < t + h | X > t } = μh + o ( h ) as h → 0. Here o ( h ) is a function of h s.t. lim h → o ( h ) h = 0 . Solution: Suppose X has an exponential distribution, P { X < t + h | X > t } = 1- e- μh (by lack of memory property) (Use Taylor’s series) = 1- (1- μh + o ( h )) = μh + o ( h ) as h → ....
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midterm2_Solution_Prob_Thy - MATHEMATICS 2603 Midterm,...

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