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(3) Let 7; be the stationary distribution at webpage i, so that ; = Σ₁=₁ #iKij. Let π = (ni, i = 1,2,3) be the row vector. Then = K. Given K, solve from this equation. Is p(3) close to ? (4) Based on the above calculations, answer the following questions. Suppose there are 1 million people doing the above random walk independently, and suppose they all start from webpage 1 at time t = 0. Then on average, what is the distribution of these 1 million people for t = 1,2,3? What is the stationary distribution of these 1 million people? Which page is the most popular?

(3) Let 7; be the stationary distribution at webpage i, so that ; = Σ₁=₁ #iKij. Let π = (ni, i = 1,2,3) be the row vector. Then = K. Given K, solve from this equation. Is p(3) close to ? (4) Based on the above calculations, answer the following questions. Suppose there are 1 million people doing the above random walk independently, and suppose they all start from webpage 1 at time t = 0. Then on average, what is the distribution of these 1 million people for t = 1,2,3? What is the stationary distribution of these 1 million people? Which page is the most popular?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Please solve part 3 and 4

Problem 4 Consider a random walk over 3 webpages, 1, 2, 3. At any step, if the person is at webpage 1, then with probability 1/6, she will go to webpage 2, and with probability 1/6, she will go to webpage 3. If the person is at webpage 2, then with probability 1/2, she will go to webpage 1, and with probability 1/2, she will go to webpage 3. If the person is at webpage 3, then with probability 1/2, she will go to webpage 1, and with probability 1/2, she will go to webpage 2. Let X be the webpage the person is browsing at time t, and let us assume she starts from webpage 1 at time 0, i.e., Xo = 1. (1) Let Kij = P(Xt+1 = j|Xt = i). Let K = K. = (2) Let p D(t) P(Xi). Let p(t) t = 1, 2, 3 using vector matrix multiplication. = (3) Let 7; be the stationary distribution at webpage i, so that j = Σ₁-₁ T₁K₁j. Let π = = (πi, i 1,2,3) be the row vector. Then = TK. Given K, solve from this equation. Is p(3) close to π? (Kij) be the 3 x 3 transition matrix. Write down (p),i=1,2,3) be the row vector. Calculate p(t) for = (4) Based on the above calculations, answer the following questions. Suppose there are 1 million people doing the above random walk independently, and suppose they all start from webpage 1 at time t = 0. Then on average, what is the distribution of these 1 million people for t = 1,2,3? What is the stationary distribution of these 1 million people? Which page is the most popular?
Transcribed Image Text:Problem 4 Consider a random walk over 3 webpages, 1, 2, 3. At any step, if the person is at webpage 1, then with probability 1/6, she will go to webpage 2, and with probability 1/6, she will go to webpage 3. If the person is at webpage 2, then with probability 1/2, she will go to webpage 1, and with probability 1/2, she will go to webpage 3. If the person is at webpage 3, then with probability 1/2, she will go to webpage 1, and with probability 1/2, she will go to webpage 2. Let X be the webpage the person is browsing at time t, and let us assume she starts from webpage 1 at time 0, i.e., Xo = 1. (1) Let Kij = P(Xt+1 = j|Xt = i). Let K = K. = (2) Let p D(t) P(Xi). Let p(t) t = 1, 2, 3 using vector matrix multiplication. = (3) Let 7; be the stationary distribution at webpage i, so that j = Σ₁-₁ T₁K₁j. Let π = = (πi, i 1,2,3) be the row vector. Then = TK. Given K, solve from this equation. Is p(3) close to π? (Kij) be the 3 x 3 transition matrix. Write down (p),i=1,2,3) be the row vector. Calculate p(t) for = (4) Based on the above calculations, answer the following questions. Suppose there are 1 million people doing the above random walk independently, and suppose they all start from webpage 1 at time t = 0. Then on average, what is the distribution of these 1 million people for t = 1,2,3? What is the stationary distribution of these 1 million people? Which page is the most popular?
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