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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 Â¡ P = & & & 1 Â¡ and that & + & 1 = 1 b. Obviously f 1 00 = P 00 = 1 & Â¡: For higher n; because the only path starting from 0 and reaching 0 after n steps without passing 0 is 01 Â¡ Â¡ Â¡ 10 ; its probability is f ( n ) 00 = P 01 P n & 2 11 P 10 = Â¡Â¢ (1 & Â¢ ) n & 2 c. m = 1 X n =1 nf ( n ) 00 = (1 & Â¡ ) + 1 X n =2 nÂ¡Â¢ (1 & Â¢ ) n & 2 = (1 & Â¡ ) & Â¡Â¢ 1 & Â¢ + 1 X n =1 nÂ¡Â¢ (1 & Â¢ ) n & 2 = (1 & Â¡ ) & Â¡Â¢ 1 & Â¢ + 1 X n =1 1 X k = n Â¡Â¢ (1 & Â¢ ) k & 2 = (1 & Â¡ ) & Â¡Â¢ 1 & Â¢ + 1 X n =1 Â¡ (1 & Â¢ ) n & 2 = (1 & Â¡ ) & Â¡Â¢ 1 & Â¢ + Â¡ (1 & Â¢ ) Â¢ = Â¡ + Â¢ Â¢ = 1 & 2 2 Problem 4.5 a. m ij = 1 + 0 Â¢ 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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PS4solution - Problem Set 4 Solutions 1 1 Problem 4.1 a...

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