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ex3soln

# ex3soln - ECE320 Solutions to Third Examination Spring 2006...

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ECE320 Solutions to Third Examination Spring 2006 Cornell University T.L.Fine 1. A finite state system with certain random inputs moves between its states such that its motion can be described by a Markov chain with the one-step transition probability matrix P = . 6 . 4 0 0 0 0 . 9 . 1 0 0 0 0 . 5 . 5 0 0 0 . 6 . 4 0 0 0 0 0 1 . (a)(3pts) Sketch a state transition diagram for P . See Figure 1. 1 2 3 4 5 1 .4 .6 .1 .9 .5 .5 .4 .6 Figure 1: (b)(5pts) Identify all communicating classes of states and determine whether they are open or closed. C 1 = { 1 } , open C 2 = { 2 } , open C 3 = { 3 , 4 } , closed C 4 = { 5 } , closed. 1

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(c)(4pts) If the Markov chain state X 1 is equally probable to occupy any of its states, what are the probabilities for state occupancies of X 2 ? The probabilities π (2) = π (1) P = [ . 2 . 2 . 2 . 2 . 2] P = 1 5 [ . 6 1 . 3 1 . 2 . 9 1] . (d)(3pts) If the Markov chain starts with X 1 = 3, what are the long-run probabilities of the chain occupying its states? If we start in state 3 then we remain in the closed class C 3 = { 3 , 4 } . The one-step transition matrix for
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