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STA3007_0506_t07

# STA3007_0506_t07 - STA 3007 Applied Probability Tutorial 7...

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STA 3007 Applied Probability 2005 Tutorial 7 1. Revision (a) Review on Probability (b) Introduction to Markov Chain (c) First Step Analysis i. Probability of absorption in k, given an initial state i U ik = P { X T = k | X 0 = i } ii. Expected absorption time given an inital transient state i v i = E [ T | X 0 = i ] , i = 0 , 1 , 2 , ..., r - 1 iii. For a transition matrix P = Q R 0 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 ] = R + QU = ( I - Q ) - 1 R , where U k = ( U 0k , U 1k , ..., U r - 1 , k ) and V = ( v 0 , v 1 , ..., v r - 1 ) = [ 1 ] + QV = ( I - Q ) - 1 [ 1 ] (d) The Long Run Behavior of Markov Chains i. Regular Transition Probability Matrices ii. Limiting Probability Distribution For a Markov Matrix P ij , the limiting distribution is defined as lim n →∞ P ( n ) ij = π j > 0 , for j = 0 , 1 , · · · , N Let P be a Regular Markov matrix on the states 0 , 1 , · · · , N . Then the limiting distribu- tion π = ( π 0 , π 1 , · · · , π N ) is the unique nonnegative solution of the equations π j = N k =0 π k P kj , j = 0 , 1 , · · · , N N k =0 π k = 1 iii. The Classification of States

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