Transcription of Chapter 8: Markov Chains - Auckland
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Chapter 8: Markov Chains - Auckland
149 Chapter 8: Markov far, we have examined several stochastic processes usingtransition diagrams and First-Step processes can be written as{X0, X1, X2, ..},whereXtis thestate at the transition diagram,Xtcorresponds towhich box we are in at the Gambler s Ruin (Section ),Xtis the amount of money the gamblerpossesses after tosst. In the model for gene spread (Section ),Xtis thenumber of animals possessing the harmful allele A in processes that we have looked at via the transition diagram have a crucialproperty in common:Xt+1depends only does notdepend uponX0, X1, .. , Xt like this are calledMarkov :Random Walk (see Chapter 4)time tnone of these steps matter for time t+1??time t+1In a Markov chain , thefuture depends onlyupon the present:NOT upon the text-book imageof a Markov chain hasa flea hopping about atrandom on the verticesof the transition diagram,according to the probabilities transition diagram above shows a system with 7 possible states:state spaceS={1,2,3,4,5,6,7}.
The matrix describing the Markov chain is called the transition matrix. It is the most important tool for analysing Markov chains. Transition Matrix list all states X t list all states z }| {X t+1 insert probabilities p ij rows add to 1 rows add to 1 The transition matrix is …
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