Question

Can you help me with the following MARKOV CHAINS problem, explaining the solution step by step
Families in a certain country are classified according to whether they reside in rural, urban or suburban areas. Demographic mobility studies estimate that, on average, over the course of a year, 15% of urban families change residence and move to a suburban area, and 5% to a rural area; while 6% of families residing in suburban areas move to urban areas, and 4% to rural areas, and finally 4% of rural families migrate to urban areas and 6% to suburban areas.
a. Define states and transition matrix
b. What is the probability that a family that now lives in an urban area will continue to live in an urban area within two years? And in a suburban? And in a rural one?
c. Suppose that presently 40% of the country's families live in urban areas, 35% in suburban areas, and 25% in rural areas. What percentage of families will live in urban areas in two years?
d. What population distribution should be anticipated in the future if trends do not change?

Answer 1

a. States: 1- Living in urban area, 2 – Living in suburban area 3 – Living in rural area (1+1) (1 + 2) (1 +3)] 0.8 0.15 0.05 Transition matrix := p(1) (2 + 1) (2 + 2) ( 23) 0.06 0.9 0.04 (3+1) (3 + 2) (3+3) 0.04 0.06 0.9

So ans : 6. We want 2 - step transition matrix p(2) [0.6510 0.2580 0.091 p(2) p(1) p(1) 0.1036 0.8214 0.075 Pl? 0.6510 0.0716 0.114 0.8144 urban to urban in 2 years, P2 = 0.2580 urban to suburb in 2 yrs, P = 0.091 = urban to rural in 2 yrs.

$c.\ We\ know\ current\ distribution\ a^{(0)}=[0.4, \ 0.35, \ 0.25]. We\ want\ distribution\\ after\ 2\ years. \\ a^{(2)} = a^{(0)}P^{(2)}=\begin{bmatrix} 0.4 & 0.35 & 0.25 \end{bmatrix} \begin{bmatrix} 0.6510 &0.2580 & 0.091 \\ 0.1036 & 0.8214 & 0.075 \\ 0.0716 & 0.114 & 0.8144 \end{bmatrix} = [0.31456, \ 0.41919, \ 0.26625] \\ so\ 31.456\%\ of\ families\ will\ live\ in\ urban\ areas\ in\ two\ years.$

$d. \ let\ future\ steady\ state\ distribution\ be\ \pi_{1}, \pi_{2}, \ \pi_{3} \\ then \ , \ \pi_{1}+\pi_{2}+\pi_{3}= 1 \\ also, \ \pi_{1}=0.8\pi_{1} + 0.06\pi_{2} + 0.04\pi_{3} \\ \pi_{2} = 0.15\pi_{1} + 0.9\pi_{2} + 0.06\pi_{3} \\ solving\ above\ 3\ eqns\ we\ get, \\ \pi_{1} = 0.25969 , \ \pi_{2}=0.36431 , \ \pi_{3}=0.376$