1 answer

Can you help me with the following MARKOV CHAINS problem, explaining the solution step by step...

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?


Answers

a. \ States: \ 1-Living\ in\ urban\ area, \ 2-Living\ in\ suburban\ area\\ 3-Living\ in\ rural\ area \\ Transition\ matrix := \ P^{(1)}=\begin{bmatrix} (1\to1) & (1\to 2) & (1\to 3) \\ (2\to 1) &(2\to2) & (2\to3) \\ (3\to1) & (3\to2) & (3\to3) \end{bmatrix} = \begin{bmatrix} 0.8 & 0.15 & 0.05 \\ 0.06 & 0.9 & 0.04 \\ 0.04 & 0.06 & 0.9 \end{bmatrix}

b.\ We\ want\ 2-step\ transition\ matrix\ P^{(2)}\ . \\ P^{(2)}=P^{(1)}P^{(1)} = \begin{bmatrix} 0.6510 &0.2580 & 0.091 \\ 0.1036 & 0.8214 & 0.075 \\ 0.0716 & 0.114 & 0.8144 \end{bmatrix} , \ \ so\ ans : P^{(2)}_{11}=0.6510=urban\ to\ urban\ in\ 2\ years, \\ P^{(2)}_{12}=0.2580=urban\ to\ suburb\ in\ 2\ yrs , \ \ \ P^{(2)}_{13}=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.</p><p>\ 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

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