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EE465-USC-Fall08-Assignment5 - game without slipping back...

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EE 465 : Homework 5 Due : 10/9/2008, Thursday in class. 1. Solve 4.2 (4.2) from textbook. (Hint: Define a state as what happened in the last three days) 2. Solve 4.4 (4.4) from textbook. (Suggestion: Try to solve the question before looking at the solutions on the back of the book) 3. Solve 4.6 (4.6) from textbook. 4. Solve 4.8 (4.8) from textbook. 5. Solve 4.14 (4.14) from textbook. 6. Solve 4.18 (4.18) from textbook. 7. Solve 4.20 (4.20) from textbook. 8. A simple game of snakes and ladders is played on a board of nine squares. At each turn a player tosses a fair coin and advances one or two places according to whether the coin lands heads or tails. If you land at the foot of a ladder you climb to the top, but if you land at the head of a snake you slide down the tail. a) How many turns on the average does it take to complete the game? b) What is the probability that a player who has reached the middle square will complete the
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Unformatted text preview: game without slipping back to square 1? (Hint: Model the game as a Markov Chain, and calculate μ 19 .) (Superhint: Do you really need 9 states in your model?) Problem 9 is an optional question, those who solve it will get extra credits. 9. Let P n × n be the transition matrix of a markov chain with a Fnite state space S = { 1 , 2 , . . . n } . a) Show that Π 1 × n is the stationary distribution of the Markov chain if and only if Π ( I-P + A ) = a , where A = ( a ij : i, j ∈ S ) with a ij = 1 for all i and j . a 1 × n = ( a i : i ∈ S ) with a i = 1 for all i . and I n × n is the identity matrix. b) Show that if the Markov chain is irreducible, then ( I + P-A ) is invertible. Note that now we can compute Π by simply inverting a matrix, that is Π = a ( I-P + A )-1 Note : 4.8 (4.7) implies problem 4.8 from the ninth edition which is the same as 4.7 in the eighth edition. 2...
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EE465-USC-Fall08-Assignment5 - game without slipping back...

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