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Markov Chains: Part II

Markov Chains: Part II. We now return to the Typical Problem. A Typical Problem. The city of Math Island is experiencing a movement of its population to the suburbs. At present, 85% of the total population lives in the city and 15% lives in the suburbs.

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Markov Chains: Part II

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  1. Markov Chains: Part II

  2. We now return to the Typical Problem.

  3. A Typical Problem The city of Math Island is experiencing a movement of its population to the suburbs. At present, 85% of the total population lives in the city and 15% lives in the suburbs. But each year, 7% of the city people move to the suburbs, while only 1% of the suburb people move back to the city. Assume that the total population remains constant. Let C0, C1, C2, ⋅⋅⋅ represent the percentages of the population in the city, respectively, now, 1 year from now, 2 years from now, and so on. Also, let S0, S1, S2, ⋅⋅⋅ be the percentages of the population in the suburbs, respectively, now, 1 year from now, 2 years from now, and so on.

  4. Question #8 We found that and It is evident that the percent of the total population that remains in the city is decreasing. We wonder whether the situation ever stabilizes, i.e. after a certain number of years will an equilibrium be ever reached? Use your calculator to compute Dn for very large n values. What do you notice?

  5. Answering Question #8 Recall: The initial state matrix is and the transition matrix is Compute, say D25, D50, D100, and D200. and

  6. Answering Question #8: continued We use a calculator to perform the matrix multiplication and obtain: City Suburbs City Suburbs City Suburbs City Suburbs and It seems that in the long run 12.5% of the population remains in the city and 87.5% resides in the suburbs.

  7. Question #9 In the previous question, you found the probability distribution of the residents in the long run. Your friend, a former MA 110 student, says that the probability distribution of the residents in the long run does not depend on the initial distribution D0. Compute Mk for k=10, 50,80, 90, …, where M is the transition matrix for the Typical Problem. What do you notice?

  8. Answering Question #9 We use a calculator to raise M to the different powers. If we round to 3 decimal places, we have: and • For large values of k, note that: • The rows of the Mk are becoming identical! • Each row is approaching Dk, where

  9. Question #10 It seems that your friend is right! Let us try to understand what is happening. City Suburbs Let be the probability distribution when the population movement stabilizes, i.e. x% of the residents live in the city and y% live in the suburbs. Why are the following mathematical statements true: • x + y =1 ? (1) • D=DM ? (2) where M is the transition matrix

  10. Answering Question #10 (1) x +y gives 100% of the population of Math Island. So we must have (2) The population is assumed to be stabilized. So there should no change or no significant change once the population distribution has reached D. Remark: In Markov chain theory, D is called a stationary matrix.

  11. Question #11 Find the stationary matrix for the Typical Problem. Hint: What systems of equations do we need to solve?

  12. Answering Question #11 The stationary matrix satisfies: D = DM, where and By matrix equality, we have:

  13. Answering Question #11: Continued The last 2 equations are equivalent to But we know that So we solve the system: The solution is and Indeed, in the long run 12.5% of the population remains in the city and 87.5% resides in the suburbs.

  14. We now apply what we’ve learned from the Typical Problem to solve a Markov chain problem

  15. Example The buying pattern of Lost City home buyers who buy single-family homes and condominiums has been observed and it was discovered that: 85% of single-family homeowners buy again single-family homes and 65% of condominium owners buy again condominiums. If this trend continues, in the long run what will be the percentage of homeowners in the city that will own single-family homes and condominiums?

  16. SFH Condos SFH Condos Let T be the transition matrix. Then we have If S is the stationary matrix, where We have: and Solving the system that we obtain, as before: and

  17. Comments • We considered Markov processes with two states. We can generalize the same steps to a 3-state Markov chain. • We’d have to deal with a 1 X 3 initial state matrix and 3 X 3 transition matrix. • If the Markov chain is what we call a regular Markov chain, we can obtain its stationary matrix by raising the transition matrix to higher and higher powers until the rows become identical. • This can be a substitute for solving the system that we usually solve to find the stationary matrix.

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