Problem%20Set%201

# Problem%20Set%201 - STAT 433/833 Problem Set 1 1 Exercise...

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STAT 433/833 Problem Set 1 1. Exercise 1.7, pg. 36 of the textbook. 2. Exercise 2.2, pg. 57 of the textbook. 3. Exercise 2.4, pg. 58 of the textbook. 4. Exercise 2.5, pg. 58 of the textbook. 5. Exercise 2.7, pg. 59 of the textbook. 6. Suppose that the number of states in a Markov chain (MC) is N < . Prove that if state j is accessible from state i , then state j can be reached in N steps or less. 7. Prove that a finite-state MC has at least one recurrent state. 8. The state space S of a discrete-time MC { X n , n = 0 , 1 , 2 , . . . } is the set of all non-positive integers, namely S = { 0 , - 1 , - 2 , . . . } . For each i S , the one-step transition probabilities have the form P i,i - 1 = α 2 - i and P i, 0 = 2 - i - α 2 - i . (a) State any restrictions on the value of α that are required for P = [ P i,j ] to be a transition probability matrix (TPM) when defined as above. (b) Let N = min { n Z + : X n = 0 } be the time until the first return to state 0. For each value of α , satisfying the restrictions of part (a), find an expression for the probability mass function (PMF) of N , conditional on X 0 = 0. (c) For each value of α , satisfying the restrictions of part (a), discuss the transience and recurrence of each state of S . For those states which are recurrent, distinguish between null and positive recurrence. (d) For what values of α does the MC have a limiting distribution on S which is independent of the starting state? Justify your response.

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• Fall '13
• Probability theory, Markov chain, Xn, discrete-time MC

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