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

Question

Each American family is classified as living in an urban, rural, or suburban location. During a given year, 5% of all urban families move to a suburban location, and 15% move to a rural location; also, 4% of all suburban families move to an urban location, and 6% move to a rural location; finally, 6% of all rural families move to an urban location, and 4% move to a suburban location (a) If a family now lives in an urban location, what is the probability that it will live in an urban area two vears from now? A suburban area? A rural area? (b) Suppose that at present, 35% of all families live in an urban area, 40% live in a suburban area, and 25% live in a rural area. Two years from now, what percentage of American families will live in an urban area? (c) What problems might occur if this model were used to predict the future population distribution of the United States?

Explanation / Answer

1 - urban ,
2 - suburban
3 - rural

transition matrix
P = [0.8 0.05 0.15;0.04 0.9 0.06;0.06 0.04 0.9]
P =

    0.8000    0.0500    0.1500
    0.0400    0.9000    0.0600
    0.0600    0.0400    0.9000

a)
P^2
ans =

    0.6510    0.0910    0.2580
    0.0716    0.8144    0.1140
    0.1036    0.0750    0.8214

hence
if starting in urban
then in two year
1) urban = 0.651
2) suburban area = 0.0910
3) rural area = 0.258

b)
pi=[0.35 0.40 0.25]
pi * P^2

ans =

    0.2824    0.3764    0.3412
hence
28.24 % will stay in urban

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