MarkovChainsBianchiModel

# - Markov Chains and Bianchi Model Ydo Wexler Dan Geiger Markov Process Markov Prope Thestateof thesystemat timet 1 depends only on thestateof the

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. Markov Chains and Bianchi Model © Ydo Wexler & Dan Geiger

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2 Markov Process Markov Property: The state of the system at time t +1 depends only on the state of the system at time t X 1 X 2 X 3 X 4 X 5 [ ] [ ] x | X x X x x X | X x X t t t t t t t t = = = = = + + + + 1 1 1 1 1 1 Pr Pr Stationary Assumption: Transition probabilities are independent of time ( t ) [ ] ab t t p b a | X X = = = + 1 Pr
3 Weather: raining today 40% rain tomorrow 60% no rain tomorrow not raining today 20% rain tomorrow 80% no rain tomorrow Markov Process Simple Example rain no rain 0.6 0.4 0.8 0.2 Stochastic FSM:

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4 Weather: raining today 40% rain tomorrow 60% no rain tomorrow not raining today 20% rain tomorrow 80% no rain tomorrow Markov Process Simple Example = 8 . 0 2 . 0 6 . 0 4 . 0 P Stochastic matrix: Rows sum up to 1 Double stochastic matrix: Rows and columns sum up to 1 The transition matrix:
5 Given that a person’s last cola purchase was Coke , there is a 90% chance that his next cola purchase will also be . If a person’s last cola purchase was Pepsi , there is an 80% chance that his next cola purchase will also be . coke pepsi 0.1 0.9 0.8 0.2 Markov Process Coke vs. Pepsi Example = 8 . 0 2 . 0 1 . 0 9 . 0 P transition matrix:

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6 Given that a person is currently a Pepsi purchaser, what is the probability that he will purchase Coke two purchases from now? Pr [ ? Coke ] = Pr + Pr = 0.2 * 0.9 + 0.8 * 0.2 = 0.34 = = 66 . 0 34 . 0
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## This note was uploaded on 11/03/2009 for the course COMPUTERS CS537 taught by Professor Salman during the Spring '09 term at Texas A&M University–Commerce.

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- Markov Chains and Bianchi Model Ydo Wexler Dan Geiger Markov Process Markov Prope Thestateof thesystemat timet 1 depends only on thestateof the

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