A Markov chain has transition matrix글 031Given the initial probabilities ø1 = $2 = ¢3 = , find Pr (X1 # X2).13 112 ㅇ

Given the transition matrix of the Markov chain is P=1216131201234140 The initial probabilities…

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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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11. A certain mobile phone app is becoming popular in a large population. Every week 10% of those who are not using the app, either because they don't have it yet or have it but are not using it, start using it, and 15% of those who are using it stop using it. Assume that the starting percentages are that 80% are not using it and 20% are using it. (a) Show the Markov matrix A representing the situation. (Letting .80 represent the starting 1.20 .80 percentages, remember that you want the situation after one week to be given by A .) .20 (b) What percentage will be using the app after two weeks? (c) Find the eigenvalues and eigenvectors of A. (d) Show a matrix X which diagonalizes A by means of X-'AX. (d) In the long run the percentages not using the app will be Show your work. _% and using it will be _%
7
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Consider a random walk on the graph with 6 vertices below. Suppose that 0 and 5 are absorbing vertices, but the random walker is more strongly attracted to 5 than to 0. So for each turn, the probability that the walker moves right is 0.7, while the probability he moves left is only 0.3. (a) Write the transition matrix P for this Markov process. (b) Find POO. (c) What is the probability that a walker starting at vertex 1 is absorbed by vertex 0? (d) What is the probability that a walker starting at vertex 1 is absorbed by vertex 5? (e) What is the probability that a walker starting at vertex 4 is absorbed by vertex 5? (f) What is the probability that a walker starting at vertex 3 is absorbed by vertex 5? What is the expected number of times that a walker starting at vertex 1 will visit vertex 2? (h) What is the expected number of times that a walker starting at vertex 1 will visit vertex 4? N;B The diagram is a horizontal line showing points 0, 1, 2, 3, 4, and 5
A square matrix is said to be doubly stochastic if itsentries are all nonnegative and the entries in each row andeach column sum to 1. For any ergodic, doubly stochasticmatrix, show that all states have the same steady-stateprobability.
At Suburban Community College, 30% of all business majors switched to another major the next semester, while the remaining 70% continued as business majors. Of all non-business majors, 10% switched to a business major the following semester, while the rest did not. Set up these data as a Markov transition matrix. HINT [See Example 1.] (Let 1 business majors, and 2 = non-business majors.) = Calculate the probability that a business major will no longer be a business major in two semesters' time.

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A Markov chain has transition matrix 글 0 글 3 Given the initial probabilities ø1 = $2 = $3 = , find Pr (X1 # X2). %3D