In a survey, it was found that $69 %$ of the population of a certain urban area lived in single-family dwellings and $31 %$ in multiple housing. Five years later, of those who had been living in single-family dwellings, $86 %$ still did so, but $14 %$ had moved to multiple-family dwellings. Of those in multiple housing, $95 %$ were still living in that type of housing. while $5 %$ had moved to single-family housing. Assume that these trends continue. Answer parts (a) through (f).
(a) Fill in the appropriate transition matrix for this information.
single multiple
$multiple housing [\begin{array}{ll} 0.86 & 0.14 \\ 0.05 & 0.95 \end{array}]$
(Type an integer or decimal for each matrix element.)
(b) Find the probability vector for the initial distribution of housing
$x_{0} = ◻$
(Type an integer or decimal for each matrix element.)

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Thus, the transition matrix is $[\begin{array}{ll} 0.86 & 0.14 0.05 & 0.95 \end{array}]$ and the initial distribution of housing is $x_{0} = [0.69, 0.31]$ .

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Step 1 ：Given that 69% of the population lived in single-family dwellings and 31% in multiple housing, we can represent this as an initial distribution vector $x_{0} = [0.69, 0.31]$ .

Step 2 ：Also, we know that of those who had been living in single-family dwellings, 86% still did so, but 14% had moved to multiple-family dwellings. Of those in multiple housing, 95% were still living in that type of housing, while 5% had moved to single-family housing. This can be represented as a transition matrix $T = [[0.86, 0.14], [0.05, 0.95]]$ .

Step 3 ：Thus, the transition matrix is $[\begin{array}{ll} 0.86 & 0.14 0.05 & 0.95 \end{array}]$ and the initial distribution of housing is $x_{0} = [0.69, 0.31]$ .

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