Question

Data collected from selected major metropolitan areas in the eastern United States show that 2% of individuals living within the city limits move to the suburbs during a one-year period, while 1% of individuals living in the suburbs move to the city during a one-year period. Assuming that this process is modeled by a Markov process with two states: city and suburbs.

a. Prepare the matrix of transition probabilities.
b. Compute the steady-state probabilities.
c. In a particular metropolitan area, 40% of the population lives in the city, and 60% of the population lives in the suburbs. What population changes do your steady-state probabilities project for this metropolitan area?

Answer 1

a) City Suburbs city 0.98 0.02 , Suburbs 0.01 0.99

b) 0.333 , 0.667

c ) Using the steady-state probabilities, There will be an increase in the Suburb population and a decrease in City population

Explanation:

2% living within the city limits move to suburbs

1% living within the suburbs move to the city

a) Matrix of transition probabilities

City Suburbs city 0.98 0.02 , Suburbs 0.01 0.99

b) Steady -state probabilities

attached below

steady state probabilities = 0.333 , 0.667

c) Determine the population changes the steady-state probabilities

Using the steady-state probabilities, There will be an increase in the Suburb population and a decrease in City population i.e. a decrease from 40% to 33%