Question

Each American family is classified as living in an urban, rural, or suburban location. During a given year, 6.0% of all urban families move to a suburban location, and 12.0% move to a rural location; also, 4.0% of all suburban families move to an urban location, and 5.0% move to a rural location; finally, 8.0% of all rural families move to an urban location, and 6.0% 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 years from now? A suburban area? A rural area? (b) Suppose that at present, 30.0% of all families live in an urban area, 45.0% live in a suburban area, and 25.0% live in a rural area. Three 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?

Answer 1

( Solution - This problem is based on Markov chain. Let'u'be Amenican family living in urban location. 'R be American family living in Suburban "location V Ś be Arican Pamily living in Suburban O rocalich location. 1) Let us dyaw the Ivorcidion diagram ho fransition produbily malytice 1. It is given thal, 6% of the urban families move to Suburban location and 121. to rural. So, (100-6-12) - 827 of urban families remain in urban location. So, row 'u' and Column 'u'20.82 Similarly, all other entire are calculated in the same way h om The transition chagram shows all other entires The Hansition diogram and transition probabiling marrin uis cogiden og hollows'

TURIS P- 0.92 0.12 0.06 TR10.08 1 0.861 806 | 50.04 0.05 0.91 a) If the family is currently living Urban location, then the probability will live in an urban area hoo from now is P (X2=0/X0 = 0) = Poula) le p²=p*p in an that it years p2 10.82 0.12 0.06 ro.sa 0.12 0.06) 10.08 0.86 0.06x Joos 886 0.06 (0.04 0.05 0.91 / 0.04 0.05 0.91 -10.6844 0.2046 a.lll 7 10.7368 0.7522 0.lll 10.0732 0.0933 0.9335 Hence, the probability that it will live in an urban area two years from now is poula) -0.6844 - Similarly, the probablity that a family living in urban area will live in a suburban area two years from now is Pus (2) = 0.1l.

Tavec Vyuya - And, probability that a family living e in urban area will live a a two years from now is PUR (a) = 0,2046. = b) Let us assume that initial state matrix 1. from the question we have, 30% of all a families live in an urban area, 45%. 2 live in Suburban and 25%. live in rural area - Thus = 9103 0.25 0.45 1. The percentage of American families who a lives in an voban, rural and Suburban area - three years from now us 1. The product of matris g and the • Column 1 of pe • So led us hird pe now 10.6844 0.2046 o.llll 10. 1368 0.7522 0.111 p3. P*P= | 0.0132 0.0933 0.8335 0.82 0.08 0.04 0.12 0.06 0.86 0.06 0.95 0.91

0.154441 0.154441 0.7684737 -0.58202 0263639 10.176796 0.668863 0.100828 0.130697 s*p3 - (0.3 6.25 6.457* 0.58202 0.263639 0154441 0.154441 0.768475 6.176796 0.100828 0.668863 0.130697 = [0.2642 0.3051 8.4307] Hence from the above matrine, loe can say that 26.42% of population of American families will live in an urban area, 30.51%. will live in rural area and 43.07% will live in suburban area abter 3 years from now. c) During the period of time, moving tende- -ncies of pepole change. Long back. population in rural area was in decline stage since people move to find jobs in urban area. Gradually, they started moving too Subura ban area of rural area

practically lith on we cant predict the probah. chance in time and people's so there is a non- stationary lities with chance in to decision too. so there is a Moskov chain here. Thus, this model a cannot be used to predict the future Doculation distribution of united state