Question

Each American family is classified as living in an urban, suburban, or rural location.  During a given year, 15% of all urban families move to a suburban location, and 5% move to a rural location; also, 6% of all suburban families move to an urban location, and 4% move to a rural location; finally, 4% of all rural families move to an urban location, and 6% move to a suburban location.

Assuming that the probability that any given family moves in any year depends only on their current location, model the movement of American families as a Markov chain and answer the following questions.

a) What are the states of Markov chain?

b) Determine the one-step transition matrix, P, for the Markov chain.

c) If a family now lives in an urban location, what is the probability that it will live in an urban location two years from now? A suburban location? A rural location?

d) Suppose that at present, 40% of all American families live in an urban area, 35% live in a suburban location, and 25% live in a rural location. Two years from now, what percentage of American families will live in an urban location?

e) If a family now lives in an urban location, what is the probability that it will live in an urban location 70years from now? A suburban location? A rural location?

f) Suppose that at present, 40% of all American families live in an urban area, 35% live in a suburban location, and 25% live in a rural location. Seventy years from now, what percentage of American families will live in

1. an urban location?

2. a suburban location?

3. a rural location?

g) Repeat part f) but suppose that currently American families are evenly distributed among the three types of locations.

Answer 1

Ans. a) The following are the states of Markov chain:

1 Urban location,

2 rural location

3 suburban location

Ans. b) All entries in the first row of  transition matrix have been spefified and similarly all other entries can be specified and thus complete  transition matrix may be obtained

Ans. c) As given 15% of all urban families move to a suburban location, and 5% move to a rural location. Thus, 20% of urban families lived in urban locations. Hence, there is a probability of 0.80 to live in an urban.

Ans. d) 31.5% of American families will live in an urban location.

Ans. e) 21% of  family will live in suburban location and 13% will live in a rural location

Ans. f) 0.651 of family will live in an urban location,0.258 of family will live in a suburban location and 0.091 of family will live in a rural location

Ans. g) If American families are evenly distributed among the three types of locations than the situation will be different.