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Unformatted text preview: ORIE3510 Introduction to Engineering Stochastic Processes Spring 2010 Midterm Review Session DTMCs • DTMC completely determined by its transition matrix P (along with an initial distribution for X ). • Always work with conditional probabilities: 1-step transition probabilities P ( X 1 = j | X = i ) given by entry P ij in transition matrix, n-step transition probabilities P ( X n = j | X = i ) given by ij-th entry in matrix P n . • Markov property: P ( X n +1 = j | X n = i,...,X = i ) = P ( X n +1 = j | X n = i ). If we know the present state, the past does not give any extra information about the future. In some cases, next step is found by independently updating the current configuration, eg. balls in urns, card ordering (cf. HW 2 Problem 5). • Communicating classes — find by drawing state space diagram. Irreducible if all states in the same class (for example if all entries in P non-zero). Class properties: transience, recurrence, null recurrence, periodicity. • Transient states: positive probability of never returning to i , starting from i . Expected return time starting in i is infinite. On diagram, if there is a path leaving i and never returning then i is transient. Less clear in infinite state space (eg. random walk withis transient....
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- Spring '09
- Probability theory, #, Markov chain, aii, sij, QII