Hw2_ORIE3510_S10_slns

# Hw2_ORIE3510_S10_slns - ORIE3510 Introduction to...

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ORIE3510 Introduction to Engineering Stochastic Processes Spring 2010 Homework 2: Discrete Time Markov Chains (solutions) Due February 10, 2010 2:30 PM (drop box) 1. Probability that the X j = i for all j = 1 , 2 ,...,n is ( P ii ) j . Therefore the distribution of η i is (1 - P ii ) * ( P ii ) n - 1 for n > 1. (i.e. η i is geometrically distributed with parameter 1 - P ii ) 2. (a) This can be modeled as a markov chain where the state space is { (WWW), (WWL), (WLW), (WLL),(LWW), (LWL), (LLW), (LLL) } . Where W = win and L = lose. For example, (WWL), means they won lost the last game, won the previous one and won the one before that. (b) Transition matrix P = (WWW) (WWL) (WLW) (WLL) (LWW) (LWL) (LLW) (LLL) (WWW) 0.7 0.3 (WWL) 0.45 0.55 (WLW) 0.55 0.45 (WLL) 0.55 0.45 (LWW) 0.55 0.45 (LWL) 0.55 0.45 (LLW) 0.55 0.45 (LLL) 0.3 0.7 3. Note that X n +1 = Y n +1 + Y n = Y n +1 + ( Y n + Y n - 1 ) - Y n - 1 = Y n +1 + X n - Y n - 1 . Hence

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## This note was uploaded on 10/07/2010 for the course ORIE 3150 taught by Professor Callister during the Fall '08 term at Cornell.

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Hw2_ORIE3510_S10_slns - ORIE3510 Introduction to...

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