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midterm452f10

# midterm452f10 - Mathematics 452[F01 Midterm examination#1...

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Mathematics 452 [F01]: Midterm examination #1 Department of Mathematics and Statistics, University of Victoria Oct. 22, 2010 This examination paper consists of 4 problems worth a total of 22 marks. Read through carefully and choose the questions you want to answer. You need to total out 20 marks. However, feel free to answer as many questions as you want. Any points that exceed 20 will be counted as bonus marks , i.e, the exam is on 20 marks plus 2 bonus points. 1. Consider a Markov chain on a countable state space with a probability transition matrix P = [ p ij ]. Let i and j be two states such that i is not accessible from j . Show that if P ij > 0, then i is transient. [4] 2. Consider the Markov chain X n whose transition probability matrix is given by [8] P = 1 / 3 2 / 3 0 0 0 0 4 / 5 1 / 5 1 0 0 0 0 0 0 1 . (a) Draw the graph or diagram of this Markov chain and use this graph to classify its sates into recurrent and transient classes. Let T be the set of all transient states. [2] (b) Let i, j be two transient states (i.e i, j T ) and

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midterm452f10 - Mathematics 452[F01 Midterm examination#1...

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