STA3007_0506_t03

STA3007_0506_t03 - STA 3007 Applied Probability 2005...

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Unformatted text preview: STA 3007 Applied Probability 2005 Tutorial 3 1. First Step Analysis (a) Notation i. Absorbing time T ii. Probability of absorption in k, given an initial state i U ik = P { X T = k | X = i } iii. Expected absorption time given an inital transient state i v i = E [ T | X = i ] , i = 0 , 1 , 2 , ..., r- 1 iv. For a transition matrix P = Q R I 0 is an (N-r+1) by r matrix with all entries being zero. I is an (N-r+1) by (N-r+1) identity matrix. Q = [ Q ij ] where Q ij = P ij . Then U = [ U r U r + 1 ... U N ] = ( I- Q )- 1 R , where U k = ( U 0k , U 1k , ..., U r- 1 , k ) and V = ( v , v 1 , ..., v r- 1 ) = [ 1 ] + QV Exercise: i. Consider the Markov Chain whose transition probability matrix is given by: P = 1 2 3 1 . . . . 1 . 1 . 6 . 1 . 2 2 . 2 . 3 . 4 . 1 3 . . . 1 . (a) Starting in state 1, determine the probability that the Markov Chain ends in state 0. (b) Determine the mean time to absorption....
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This note was uploaded on 05/21/2011 for the course STA 3007 taught by Professor Kb during the Spring '11 term at CUHK.

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STA3007_0506_t03 - STA 3007 Applied Probability 2005...

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