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(X4+1 = j|Xt = i). Let K = (Kij) be the 3 x 3 transition matrix. Write down K. %3D %3D (2) Let p = P(X, = i). Let p = (p,i = 1,2,3) be the row vector. Calculate p) for t = 1,2,3 using vector matrix multiplication. (3) Let 7, be the stationary distribution at webpage i, so that z, Kij. Let a (7, i = 1,2,3) be the row vector. Then 7 = #K. Given K, solve a from this equation. Is p(3) close to n? (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?
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(X4+1 = j|Xt = i). Let K = (Kij) be the 3 x 3 transition matrix. Write down K. %3D %3D (2) Let p = P(X, = i). Let p = (p,i = 1,2,3) be the row vector. Calculate p) for t = 1,2,3 using vector matrix multiplication. (3) Let 7, be the stationary distribution at webpage i, so that z, Kij. Let a (7, i = 1,2,3) be the row vector. Then 7 = #K. Given K, solve a from this equation. Is p(3) close to n? (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
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text: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(X++1 = j|X = i). Let K = (Kj) be the 3 x 3 transition matrix. Write down
%3D
К.
(t)
(2) Let p = P(Xt = i). Let p) = (p",i = 1,2,3) be the row vector. Calculate p for
t = 1,2,3 using vector matrix multiplication.
(3) Let 7; be the stationary distribution at webpage i, so that 7, = E n;Kij. Let n = (Ti, i =
1, 2, 3) be the row vector. Then 7 = aK. Given K, solve a from this equation. Is p3) close to n?
(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?
%3D
%3D
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