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A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n= 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 =2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.

A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n= 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 =2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.

A First Course in Probability (10th Edition)
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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A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n = 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 = 2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.
Transcribed Image Text:A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n = 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 = 2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.
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