Question

The city of Eden completed a new light rail system to bring commuters and shoppers into the downtown area and relieve highway congestion. City planners estimate that each year, 15% of those who drive or ride in an automobile will change to the light rail system; 80% will continue to use automobiles; and the rest will no longer go to the downtown area. Of those who use light rail; 5% will go back to using an automobile, 80% will continue to use light rail, and the rest will stay out of the downtown area. Assume those who do not go downtown will continue to stay out of the downtown area. The transition matrix associated with this Markov chain is a.an absorbing stochastic matrix. b.a regular stochastic matrix.

Answer 1

Explanation:

To check which option is correct we will first make transition matrix.

Let the state 1 and state 2 represent the automobile and light rail

So, the transition is given by

`P=\left[\begin{array}{ccc}0.80+0.05&0.15\\0.5+0.15&0.80\end{array}\right]`

`P=\left[\begin{array}{ccc}0.85&0.15\\0.20&0.80\end{array}\right]`

Since P as no any entry zero so, P as not absorbing stochastic matrix

Also,

`|P| = \left|\begin{array}{ccc}0.85&0.15\\0.20&0.80\end{array}\right| \\\\=0.85*0.8-0.15*0.2\\\\=0.65\\eq0`

since |P| ≠ 0

Hence, Markov chain is a regular stochastic matrix.