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Problem 1

Determine whether the matrix is an absorbing stochastic matrix. \(\left[\begin{array}{ll}\frac{2}{5} & 0 \\ \frac{3}{5} & 1\end{array}\right]\)

Expert verified

The given matrix \(\left[\begin{array}{ll}\frac{2}{5} & 0 \\\ \frac{3}{5} & 1\end{array}\right]\) is a square matrix that has non-negative elements. However, the sum of each row is not equal to 1 (\(\frac{2}{5}\) for row 1 and \(\frac{8}{5}\) for row 2). As it does not fulfill the conditions to be a stochastic matrix, it cannot be an absorbing stochastic matrix.

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Chapter 9

Determine whether the matrix is an absorbing stochastic matrix. $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & .7 & .2 \\ 0 & .3 & .8\end{array}\right]$

Chapter 9

Within a large metropolitan area, \(20 \%\) of the commuters currently use the public transportation system, whereas the remaining \(80 \%\) commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 mo from now \(30 \%\) of those who are now commuting to work via automobile will switch to public transportation, and \(70 \%\) will continue to commute via automobile. At the same time, it is expected that \(20 \%\) of those now using public transportation will commute via automobile and \(80 \%\) will continue to use public transportation. a. Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters. b. Find the initial distribution vector for this Markov chain. c. What percentage of the commuters are expected to use public transportation \(6 \mathrm{mo}\) from now?

Chapter 9

Brady's, a conventional department store, and ValueMart, a discount department store, are each considering opening new stores at one of two possible sites: the Civic Center and North Shore Plaza. The strategies available to the management of each store are given in the following payoff matrix, where each entry represents the amounts (in hundreds of thousands of dollars) either gained or lost by one business from or to the other as a result of the sites selected. a. Show that the game is strictly determined. b. What is the value of the game? c. Determine the best strategy for the management of each store (that is, determine the ideal locations for each store).

Chapter 9

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). $\left[\begin{array}{lll}0 & .2 & 0 \\ .5 & .4 & 0 \\ .5 & .4 & 1\end{array}\right]$

Chapter 9

Over the years, consumers are turning more and more to newer and much improved videorecording devices. The following transition matrix describes the Markov chain associated with this process. Here \(\mathrm{V}\) stands for VHS recorders, D stands for DVD recorders, and \(\mathrm{H}\) stands for high definition video recorders. $A=\begin{array}{l}\mathrm{V} \\ \mathrm{D} \\\ \mathrm{H}\end{array}\left[\begin{array}{rrr}.10 & 0 & 0 \\ .70 & .60 & 0 \\\ .20 & .40 & 1\end{array}\right]$ a. Show that \(A\) is an absorbing stochastic matrix and rewrite it so that the absorbing state appears first. Partition the resulting matrix and identify the submatrices \(R\) and \(S\). b. Compute the steady-state matrix of \(A\) and interpret your results.

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