This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
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.
 Fall '09
 GLENTAKAHARA

Click to edit the document details