Math_303_April_2009

# Math_303_April_2009 - April 2009 Marks 1 Mathematics 303...

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April 2009 Mathematics 303 Name Page 2 of 8 pages Marks 1. Consider the Markov chain with the transition matrix 0 1 2 3 4 5 0 0 1 0 0 0 0 1 1 / 2 0 0 0 1 / 2 0 2 0 0 1 0 0 0 3 0 0 1 / 3 0 2 / 3 0 4 0 0 1 / 2 0 0 1 / 2 5 1 0 0 0 0 0 [11] (a) Find the communicating classes. Determine their periods, and whether they are transient or recurrent. [7] (b) If you start in state 1, what is the probability that after 4 steps you will be back in state 1? Continued on page 3

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April 2009 Mathematics 303 Name Page 3 of 8 pages 2. The following statements refer to a Markov chain with transition matrix P . True or false? Give brief reasons. [4] (a) If state i is recurrent and P n ij > 0 for some n , then states i and j communicate. [4] (b) Given that the process starts in a recurrent state i , the expected time to return to state i is Fnite. [4] (c) There is always at least one recurrent class. Continued on page 4
April 2009 Mathematics 303 Name Page 4 of 8 pages 3. At the end of each month, a certain business Fres whichever member of its N -person sales sta± has made the least sales during the month, and replaces him or her with someone new. Assume that each sales person is equally likely to be the one Fred. Of the N people who were employed at the beginning of the year, let X n be the number who are still there after

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Math_303_April_2009 - April 2009 Marks 1 Mathematics 303...

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