Problem
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 \[ \text { multiple housing }\left[\begin{array}{ll} 0.86 & 0.14 \\ 0.05 & 0.95 \end{array}\right] \] (Type an integer or decimal for each matrix element.) (b) Find the probability vector for the initial distribution of housing \[ x_{0}=\square \] (Type an integer or decimal for each matrix element.)
Solution
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 \(\boxed{\left[\begin{array}{ll} 0.86 & 0.14 \ 0.05 & 0.95 \end{array}\right]}\) and the initial distribution of housing is \(\boxed{x_{0} = [0.69, 0.31]}\).
