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Math Help - Markov chain

  1. #1
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    Oct 2008
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    Markov chain

    Suppose three out of every four trucks on the road are followed by a car, while only one out of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?

    I form a markov chain as followed:

    \[ \left( \begin{array}{ccc}<br />
0.25 & 0.75 \\<br />
0.20 & 0.80<br />
\end{array} \right)\]

    Then find the limiting probabilities \pi_{track} and I got 0.2105.

    Am I right?
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  2. #2
    Member
    Joined
    Mar 2009
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    Yup, thats what i got, since its irreducible finite markov chain there is a stationary distribution, which is the proportion of time the chain spends in each state;
    If 0 is the event of a truck, 1 is a car (as outlined in your transition matrix) then
    \pi_0=.25\pi_0+.2\pi_1
    \pi_1=.75\pi_0+.8\pi_1 \to \pi_1=3.75\pi_0
    And \pi_0+\pi_1=1

    So solving,
    \pi_0=\frac{1}{1+3.75}=0.2105
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