Hw4-ORIE3510_S12 - ORIE 3510 Homework 4 Instructor: Mark E....

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ORIE 3510 – Homework 4 Instructor: Mark E. Lewis due 2PM, Wednesday February 22, 2012 (ORIE Hallway drop box) 1. Show that if state i is recurrent and state i does not communicate with state j , then p ij = 0. (This implies that once a process enters a recurrent class of states, it can never leave that class. For this reason, a recurrent class is often referred to as a closed class). 2. Consider a Markov chain with state space S = { 1 , 2 ,..., 9 } and transition matrix given by 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 / 2 0 0 0 1 / 2 0 0 0 0 1 / 2 0 0 0 1 / 2 0 0 0 0 0 0 0 0 0 2 / 3 1 / 3 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 / 4 0 3 / 4 0 0 0 0 0 0 0 1 0 0 0 a) Classify the states. b) Which classes are transient? Which classes are recurrent? c) What are the periods of each state? 3. A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is X i p ij = 1 for all j If such a chain is irreducible and aperiodic and consists of
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Hw4-ORIE3510_S12 - ORIE 3510 Homework 4 Instructor: Mark E....

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