We want to compute the long run average fraction of time the chain is in state

We want to compute the long run average fraction of

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We want to compute the long-run average fraction of “time” the chain is in state 0 (“Truck”). Thus, we want π 0 . The chain is ergodic (since it is irreducible and has a finite number of states, we know that each state is positive recurrent; we also check that the states have period 1 (aperiodic)). Thus, we can solve π = πP, i π i = 1 to get π 0 = (1 / 4) π 0 + (1 / 5) π 1 π 1 = (3 / 4) π 0 + (4 / 5) π 1 π 0 + π 1 = 1 , which gives π 0 = 4 / 19 (and π 1 = 15 / 19). Problem 3. (Assume that the vertices of the triangle are indexed in clockwise order around the triangle, so that vertex 2 is clockwise of vertex 1, etc.) Let X n ∈ { 1 , 2 , 3 } be the vertex where the flea is at time n. Then, we have a Markov chain with single-step transition matrix P = 0 p 1 q 1 q 2 0 p 2 p 3 q 3 0
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(a) The Markov Chain is irreducible, positive recurrent, aperiodic. We calculate the stationary distribution from π = πP, i π i = 1 : π 1 = q 2 π 2 + p 3 π 3 π 2 = p 1 π 1 + q 3 π 3 π 3 = q 1 π 1 + p 2 π 2 π 1 + π 2 + π 3 = 1 . (1) We omit any one of the first 3 equations above (since it is redundant), and solve the remaining linearly independent equations for the π i ’s. (You need not solve the equations, but you should be aware that you can solve them and you should be able to solve them!) The proportion of time the flea spends at vertex i is π i . My solution of the equations gives π 3 = 1 - p 1 q 2 2 - 2 p 1 q 2 q 3 + p 1 p 3 + q 2 q 3 π 2 = p 1 p 3 + q 3 1 - p 1 q 2 · 1 - p 1 q 2 2 - 2 p 1 q 2 q 3 + p 1 p 3 + q 2 q 3 π 1 = 1 - π 2 - π 3 .
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