midterm03sol - STAT455/855 Fall 2003 Applied Stochastic...

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Unformatted text preview: STAT455/855 Fall 2003 Applied Stochastic Processes Midterm, Brief Solutions 1. (15 marks) (a) (2 marks) When N = 1, each move from the line x = 0 will go to one of the lines x =- 1 or x = 1 with probability 1/2. Thus, the number of moves to go from the line x = 0 to one of the lines x =- 1 or x = 1 has a Geometric(1/2) distribution, with mean M (1) = 1 / (1 / 2) = 2. (b) (2 marks) Conditioning on the first move from (0 , 0) we obtain M ( N ) = (1 + M ( N ) ) 1 2 + (1 + M ( N ) 1 ) 1 4 + (1 + M ( N )- 1 ) 1 4 = (1 + M ( N ) ) 1 2 + (1 + M ( N ) 1 ) 1 2 , using observation (ii). Solving for M ( N ) , we get M ( N ) = 2 + M ( N ) 1 . (c) (7 marks) For k = 1 ,...,N- 1, we can condition on the particle’s first move starting from the line x = k to obtain M ( N ) k = (1 + M ( N ) k ) 1 2 + (1 + M ( N ) k- 1 ) 1 4 + (1 + M ( N ) k +1 ) 1 4 , noting that M ( N ) N = 0. This is equivalent to 2 M ( N ) k = 4 + M ( N ) k- 1 + M ( N ) k +1 or M ( N ) k- M ( N ) k- 1 = 4 + M ( N ) k +1- M ( N ) k . Starting from k = 1, we get M ( N ) 1- M ( N ) = 4 + M ( N ) 2- M ( N ) 1 = (4)(2) + M ( N ) 3- M ( N ) 2 . . . = (4)( N- 1) + M ( N ) N- M ( N ) N- 1 = 4( N- 1)- M ( N ) N- 1 , using M ( N ) N = 0. But, from part(b), M ( N ) 1- M ( N ) =- 2. Therefore, solving for M ( N ) N- 1 , we obtain M ( N ) N- 1 = 2 + 4( N- 1). (d) (4 marks) Starting from the line x = 0, in order to get to one of the lines x =- N or x = N we must first get to one of the lines x =- 1 or x = 1. The mean number of moves to do this is M (1) . Once we get to one of the lines x =- 1 or x = 1 we must then get to one of the lines x =- 2 or x = 2. From observation (ii), the mean number of moves to do this is M (2) 1 . Continuing this reasoning, we see that we can break up the number of moves to get from the line x = 0 to one of the lines...
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This note was uploaded on 09/15/2010 for the course STAT 455/855 taught by Professor Glentakahara during the Fall '09 term at Queens University.

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midterm03sol - STAT455/855 Fall 2003 Applied Stochastic...

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