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.