sol_assign4 - MAT1332A - Calculus for the Life Sciences II...

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Unformatted text preview: MAT1332A - Calculus for the Life Sciences II - Fall 2009 Solutions to Assignment 4 (1) In a certain region, 5% of a citys population moves to the surrounding suburbs each year, and 3% of the suburban population moves into the city. In 2008, there were 600,000 residents in the city and 400,000 in the suburbs. (a) [3 points] Find the transition matrix of the Markov chain. This matrix is also called the population migration matrix . (b) [5 points] Estimate the distribution of the population in 2009 and in 2010 ( Note: recall that the sum of the entries of the state vector must be 1 ). (c) [5 points] Find the steady-state vector (or equilibrium vector ) for the population migration matrix. Solution: (a) From : C S To : P = . 95 . 03 . 05 . 97 City Suburbs (b) Let x (0) and y (0) be the population distribution in the city and in the suburbs in 2008, respectively. Then x (1) y (1) = P x (0) y (0) = . 95 . 03 . 05 . 97 . 6 . 4 = . 582 . 418 gives the population distribution in 2009, and x (2) y (2) = P x (1) y (1) = . 95 . 03 .....
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sol_assign4 - MAT1332A - Calculus for the Life Sciences II...

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