• General notation for Markov chains: P(A) is the probability of the event A when the Markovchain starts in state x, Pμ(A) the probability when the initial state is random with distribution µ.Ty = min{n ≥ 1 : Xn = y} is the first time after 0 that the chain visits state y. px,y = Px(Ty < ∞) . Nyis the number of visits to state y after time 0.3.Each year an auto insurance company classifies its customers into three categories:Poor, Satisfactory, and Good. No one moves from poor to good or from good to poor in one year. Thestatus of a driver can be modeled by a Markov chain {Xn : n ≥ 0} with state space S = {P, S, G} andtransition matrixPS GP/1/2 1/2 0S 1/5 3/5 1/5G 0 1/5 4/5(a) Compute the invariant distribution for this Markov chain.(b) Assume that the average prices (per year) of insurance policies for drivers with Poor, Satisfactory,and Good ratings are $600, $300, $200, respectively. In the long run, how much does a driver payfor the insurance per year?(c) For a driver with a Poor rating this year, how many years does it take on average till he/she getsthe next Poor rating?(d) Find the limits of PG(Xn = S) and PG(Xn(e) For a driver with a Poor rating this year, in expectation, how many years (including this year) doeshe/she have a Poor status before receiving a Good rating?=S, Xn+2 = P) as n → ∞.

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Answered: 3. Each year an auto insurance company…

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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Aileen, a Scottish spy, has three fake identities that she uses to get information. The process is really quite involved, but she uses a Markov chain process to make it more difficult for her Irish adversaries to track her: "Hope" (state 1), "Trixie"(state 2), and "Fiona" (state 3). The transition matrix is .5 .5 P = .2 .2 .6 .4 .1 .5 On the first observation, Aileen used "Trixie" as her identity. Which of the three fake identities is she most likely to have used on the third observation? O Hope and Fiona are equally likely O Hope and Trixie are equally likely Fiona О Норе O None of the others are correct O Trixie
Long-Run Properties of Markov Chains The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst -believes that brand switching can be modeled as a Markov chain using three states, with states A and B representing customers drinking beer produced from the aforementioned breweries and state C representing all other brands. Data are taken monthly, and the analyst has constructed the following (one-step) transition -matrix from past data. What are the steady-state market shares for the two major -breweries?
A certain television show airs weekly. Each time it airs, its ratings are categorized as "high," "average," or "low," and transitions between these categorizations each week behave like a Markov chain. If the ratings are high this week, they will be high, average, or low next week with probabilities 40%, 10%, and 50% respectively. If the ratings are average this week, they will be high, average, or low next week with probabilities 60%, 20%, and 20% respectively. If the ratings are low this week, they will be high, average, or low next week with probabilities 70%, 20%, and 10% respectively. Write all answers as integers or decimals. If the ratings are average this week, what is the probability that they will be low two weeks from now? If the ratings are average the first week, what is the probability that they will be low the second week? If the ratings are low the first week, what is the probability that they will be high the third week?
Which of the following transition matrices is/are for a regular Markov Chain? 0 % 2 0 2 *0 1 1/2 X = % 0 1/2 0. Z = %3D %3D 1/2 2 0 % (A) X,Y, Z (B) Z (C) Х (D) X, Y (E) X, Z (F) none (G) Y (H) Y, Z O A OC OE O F D.
Employment Employment Last Year This Year Percentage Industry Industry 80 Small Business 10 Self-Employed 10 Small Business Industry 10 Small Business 60 Self-Employed 30 Self-Employed Industry 10 Small Business 80 Self-Employed 10 Assume that state 1 is Industry, that state 2 is Small Business, and that state 3 is Self-Employed. Find the transition matrix for this Markov process. P =
Anne and Barry take turns rolling a pair of dice, with Anne going first. Anne’s goal is to obtain a sum of 3, while Barry’s goal is to obtain a sum of 4. The game ends when either player reaches his goal,and the one reaching the goal is the winner. Define a Markov Chain to model the problem.
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In an office complex of 1000 employees, on any given day some are at work and the rest are absent. It is known that if an employee is at work today, there is an 85% chance that she will be at work tomorrow, and if the employee is absent today, there is a 70% chance that she will be absent tomorrow. Suppose that today there are 740 employees at work. (A graphing calculator is recommended.) (a) Find the transition matrix A for this scenario. at work absent A = (b) Predict the number that will be at work five days from now. (Round your answer to the nearest integer.) employees (c) Find the steady-state vector x. (Round your answer to two decimal places.)
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3. (a) A Markov process with 2 states is used to model the weather in a certain town. State 1 corresponds to a suny day. State 2 corresponds to a rainy day. The transition matrix for this Markov process is [0.7 0.4] 0.3 0.6 %3D (i) If today is rainy, what is the probability that tomorrow will be sunny? (ii) Find the steady state probability vector. (iii) In the long run, how many days a week are sunny?
The weather at a seaside resort each day is classified as either Sunny or Rainy. The probability that any two consecutive days are both sunny is 0.6, with the other three combinations being equally likely. Treating the situation as a two-state Markov chain, answer the following: (a) Find the transition matrix P and the stationary distribution vector r. (b) Find the 5-step transition matrix p(5) and hence find the probability that, if it is sunny on Sunday, it will be rainy on Friday. (c) If a day is chosen at random and found to be rainy, what is the expected number of sunny days before the next rainy day?
(c) Suppose there is an epidemic in which every month a quarter of those who are well become sick and 1/8 of those who are sick recover and acquire immunity for life. i. Find the matrix that describe this Markov process ii. Find the steady state of this process. Briefly justify your answer.

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