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.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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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