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A particle performs a random walk on the corners of the square ABCD. At each step, the probability of moving from corner c to corner d equals pcd, where PAB = PBA = PCD = PDC = α, PAD = PDA = PBC = PCB = B, and a, ß > 0, a + B = 1. Let GA (s) be the generating function of the sequence (PAA (n): n ≥ 0), where PAA (n) is the probability that the particle is at A after n steps, having started at A. Show that GA(S) = - 12 { 1 -² 3² + 1 - 18 - 0²12²5²2 1 1 1252}. Hence find the probability generating function of the time of the first return to A.

A particle performs a random walk on the corners of the square ABCD. At each step, the probability of moving from corner c to corner d equals pcd, where PAB = PBA = PCD = PDC = α, PAD = PDA = PBC = PCB = B, and a, ß > 0, a + B = 1. Let GA (s) be the generating function of the sequence (PAA (n): n ≥ 0), where PAA (n) is the probability that the particle is at A after n steps, having started at A. Show that GA(S) = - 12 { 1 -² 3² + 1 - 18 - 0²12²5²2 1 1 1252}. Hence find the probability generating function of the time of the first return to A.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 81E
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A particle performs a random walk on the corners of the square ABCD. At each step, the probability of moving from corner c to corner d equals pcd, where PAB = PBA = PCD = PDC = α, PAD = PDA = PBC = PCB = B, and a, ß > 0, a + B = 1. Let GA (s) be the generating function of the sequence (PAA (n): n ≥ 0), where PAA (n) is the probability that the particle is at A after n steps, having started at A. Show that GA(S) = - ²2 { 1 -² 3² + 1 - 18 - 0²1²25² 1 1 1252}. Hence find the probability generating function of the time of the first return to A.
Transcribed Image Text:A particle performs a random walk on the corners of the square ABCD. At each step, the probability of moving from corner c to corner d equals pcd, where PAB = PBA = PCD = PDC = α, PAD = PDA = PBC = PCB = B, and a, ß > 0, a + B = 1. Let GA (s) be the generating function of the sequence (PAA (n): n ≥ 0), where PAA (n) is the probability that the particle is at A after n steps, having started at A. Show that GA(S) = - ²2 { 1 -² 3² + 1 - 18 - 0²1²25² 1 1 1252}. Hence find the probability generating function of the time of the first return to A.
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