Question

Each American family is classified as living in an urban, rural, or suburban 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, 6% move to a suburban location.

Draw State Diagram

Construct one-step Markov Chain

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? (

Answer 1

ANSWER:-

A Markov chain can be set up from the information given, consider following states.

U: American family living in urban location

R: American family living in rural location

S: American family living in suburban location.

It is given that during a given year, 15% of urban families move to location and 5% to rural location. Thus 20% of urban families ramain in urban locations only. hence at row "U" and column "U", there is probability of 0.80 of being urban.

Also at row "U" and column "R", the probability of urban family moving to rural location is 0.05 and at row "U" and column "S", the probability of urban family moving to suburban location is 0.15.

Thus all entries in the first row of the transition matrix are defined. Similary all other entries can be defined and the complete transition matrix is obtained.

the transition probability matrix can be given as follows.

P=

	U	R	S
U	0.8	0.05	0.15
R	0.04	0.9	0.06
S	0.06	0.04	0.9

The n-step probability of transition from state i to state(P8(n)) is P8(n) = element ij of matrix P8

Let X1 be a random variable, representing the American family living in a particular location from the above transition probability.

P(X1 =U / X0 =U) = 0.8

This means that Probability that an American family living in urban location at t=1, given that at t=0 that American family is living in an urban location is 0.8.

A).

If a family is currently living in an urban location then the probability that it will be living in an urban location two years from now is

P(X2 =U / X0 =U) = PUU(2)

this is the element UU of matrix P2. Matrix P2 is obtained by matrix multiplication of P with it self that is P2 = PxP.

So

| URSURS U | 0.8 0.05 0.15 U | 0.8 0.05 0.15 R 0.04 0.9 0.06 R 0.04 0.9 0.06 $ 0.06 0.04 0.9 S 0.06 0.04 0.9 0.651 0.091 0.258 0.072 0.815 0.114 0.104 0.0751 0.821

Hence for a family currently in urban Area, the probability that the family will live in urban area 2 years from now is PUU(2) = 0.651.

Similarly, probability that a family living in urban area, will live in a suburban area 2 years from now is PUS(2) = 0.258.

Similarly, probability that a family living in urban area, will live in a rural area 2 years from now is PUR(2) = 0.091.

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