Suppose the residents of a metropolitan region move according to the probabilities in the migration matrix M in Example 1 and a resident is chosen “at random.” Then a state vector for a certain year may be interpreted as giving the probabilities that the person is a city resident or a suburban

Question

Suppose the residents of a metropolitan region move according to the probabilities in the migration matrix M in Example 1 and a resident is chosen “at random.” Then a state vector for a certain year may be interpreted as giving the probabilities that the person is a city resident or a suburban resident at that time.

a. Suppose the person chosen is a city resident now, so that [latex] x_{0} = \left [ \begin{matrix} 1 \ 0 \end{matrix} ight ] [/latex] . What is the likelihood that the person will live in the suburbs next year?

b. What is the likelihood that the person will be living in the suburbs in two years?

Accepted Answer

a. Since 5% of the city residents will move to the suburbs within one year, there is a 5% chance of choosing such a person. Without further knowledge about the person, we say that there is a 5% chance the person will move to the suburbs. This fact is contained in the second entry of the state vector [latex] x_{1} [/latex], where

[latex] x_{1} = Mx_{0} = \left [ \begin{matrix} .95 & .03 \ .05 & .97 \end{matrix} ight ] \left [ \begin{matrix} 1 \ 0 \end{matrix} ight ] = \left [ \begin{matrix} .95 \ .05 \end{matrix} ight ] [/latex]

b. The likelihood that the person will be living in the suburbs after two years is 9.6%, because

[latex] x_{2} = Mx_{1} = \left [ \begin{matrix} .95 & .03 \ .05 & .97 \end{matrix} ight ] \left [ \begin{matrix} .95 \ .05 \end{matrix} ight ] = \left [ \begin{matrix} .904 \ .096 \end{matrix} ight ] [/latex]

Question 4.9.PP.1: Suppose the residents of a metropolitan region move accordin...

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