Each American family is classified as living in an urban, suburban, or rural loc

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.

Explanation / Answer

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.

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