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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. The status of a driver can be modeled by a Markov chain {Xn n ≥ 0} with state space S = {P, S, G} and transition matrix P S G P/1/2 1/2 0 S 1/5 3/5 1/5 G 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 pay for 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 gets the next Poor rating? (d) Find the limits of PG(Xn (e) For a driver with a Poor rating this year, in expectation, how many years (including this year) does he/she have a Poor status before receiving a Good rating? = S) and PG(Xn = S, Xn+2 P) as n → ∞.

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. The status of a driver can be modeled by a Markov chain {Xn n ≥ 0} with state space S = {P, S, G} and transition matrix P S G P/1/2 1/2 0 S 1/5 3/5 1/5 G 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 pay for 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 gets the next Poor rating? (d) Find the limits of PG(Xn (e) For a driver with a Poor rating this year, in expectation, how many years (including this year) does he/she have a Poor status before receiving a Good rating? = S) and PG(Xn = S, Xn+2 P) as n → ∞.

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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• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain 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 < ∞) . Ny is 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. The status of a driver can be modeled by a Markov chain {Xn : n ≥ 0} with state space S = {P, S, G} and transition matrix PS G P/1/2 1/2 0 S 1/5 3/5 1/5 G 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 pay for 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 gets the 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) does he/she have a Poor status before receiving a Good rating? = S, Xn+2 = P) as n → ∞.
Transcribed Image Text:• General notation for Markov chains: P(A) is the probability of the event A when the Markov chain 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 < ∞) . Ny is 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. The status of a driver can be modeled by a Markov chain {Xn : n ≥ 0} with state space S = {P, S, G} and transition matrix PS G P/1/2 1/2 0 S 1/5 3/5 1/5 G 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 pay for 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 gets the 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) does he/she have a Poor status before receiving a Good rating? = S, Xn+2 = P) as n → ∞.
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