416-4sol - Denker Spring 2010 416 Stochastic Modeling -...

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Unformatted text preview: Denker Spring 2010 416 Stochastic Modeling - Assignment 4 Solution. Problem 1: Let X be a rv with values in Z and let Z n ( n 1) be independent, identically distributed rv, also independent of X , taking values in { 1;- 1 } . Prove or disprove whether X n , defined by X n +1 = X n + Z n +1 n = 0 , 1 , 2 , ..., is a Markov chain. Solution: Yes, it is a Markov chain. By independence P ( X n +1 = k | X n = j, ..., x = i ) = P ( Z n +1 = k- j, Z n = j- i n- 1 , ..., Z 1 = i 1- i , X = i ) P ( Z n = j- i n- 1 , ..., Z 1 = i 1- i , X = i ) = P ( Z 1 = k- j ) = p jk . Problem 2: (Repair shop.) During day n , Z n +1 machines break down and they enter the repair shop on day n + 1. Every day one machine waiting for service is repaired. Model the number of machines waiting for service by a Markov chain. Is the Markov chain irreducible? Draw the graph of the associated Markov chain. Solution: State space Z + = { , 1 , 2 , ... } . Let X n denote the number of defective machi- nes waiting for service on day n . Then....
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416-4sol - Denker Spring 2010 416 Stochastic Modeling -...

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