PS4solution

PS4solution - Problem Set 4 Solutions 1 1 Problem 4.1 a....

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Unformatted text preview: Problem Set 4 Solutions 1 1 Problem 4.1 a. Simply verify that & & & 1 P = & & & 1 and that & + & 1 = 1 b. Obviously f 1 00 = P 00 = 1 & : For higher n; because the only path starting from 0 and reaching 0 after n steps without passing 0 is 01 10 ; its probability is f ( n ) 00 = P 01 P n & 2 11 P 10 = (1 & ) n & 2 c. m = 1 X n =1 nf ( n ) 00 = (1 & ) + 1 X n =2 n (1 & ) n & 2 = (1 & ) & 1 & + 1 X n =1 n (1 & ) n & 2 = (1 & ) & 1 & + 1 X n =1 1 X k = n (1 & ) k & 2 = (1 & ) & 1 & + 1 X n =1 (1 & ) n & 2 = (1 & ) & 1 & + (1 & ) = + = 1 & 2 2 Problem 4.5 a. m ij = 1 + 0 Pr ( the first state is j ) + X k 6 = j E ( return time from k to j ) Pr ( the first state is k ) Pr ( the first state is k ) = P ik E ( return time from k to j j the first state is k ) = m kj Therefore m ij = 1 + X k 6 = j m kj P ik 1 b....
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This note was uploaded on 10/31/2011 for the course 18 18.445 taught by Professor Liewang during the Spring '11 term at MIT.

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PS4solution - Problem Set 4 Solutions 1 1 Problem 4.1 a....

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