Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (a)

MATH152 April 2016

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Question B 04 (a)
Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs. Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities. (a) Write down the $2\times 2$ probability transition matrix $P$ for this problem, using the ordering (1) city and (2) suburbs.

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Hint
The set of states $\{s_{1},s_{2},\cdots ,s_{n}\}$ is given and each state can move to the other states or stay the same state on each step. Then, we define transition probability $p_{ij}$ by the probability that the current state $s_{j}$ move to the state $s_{i}$ in the next step. In the question, we have two states $s_{1}=$ city and $s_{2}=$ suburb and 1 year corresponds to one step. Then, the transition probability $p_{21}$ , for example, means that $p_{21}$ percent of city residents move to the suburbs in one year.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $s_{1}=$ city and $s_{2}=$ suburbs be two states as in the hint and 1 year be one step. Since 10% of city residents move to the suburbs every year, 90% of city residents remain in the city. Using definition in the Hint, this implies that $p_{21}=0.1$ and $p_{11}=0.9$ . On the other hand, 20% of the residents of the suburbs move to cities every year, so that 80% of the suburbs residents stay in the suburbs. Then, we have $p_{12}=0.2$ and $p_{22}=0.8$ . To sum, the transition probability matrix $P=\{p_{ij}\}$ is $\color {blue}{P={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}}$ .