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Unformatted text preview: k = 1 ,X k1 = 0 ,X k2 = 0 ,. .. ,X 1 = 0  X = 1) = P 10 P k2 00 P 01 = βα k2 (1α ) 1 E ( T 1 ) = ∞ X k =1 kP ( T 1 = k ) = 1β + β ∞ X k =2 kα k2 (1α ) = 1β + β (2 + α 1α ) = 1 + β 1α 5. (a) The classes are { } , { d } , { 1,2,. . . , d1 } (b) 0 and d are the recurrent states and 1,2, .. . , d1 are the transient states. 6. (a) The only class is the whole state space. This chain is irreducible. (b) All the states are recurrent. (c) Let f be the function that relates the each state to its cost, i.e. f (0) = 20 ,f (1) = 30 ,f (2) = 50 ,f (3) = 300 E ( f ( X 1 )  X = 0) = 3 X k =0 f ( k ) P ( X 1 = k  X = 0) = 0 . 6 f (0) + 0 . 2 f (1) + 0 . 1 f (2) + 0 . 1 f (3) = 53 2...
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 Spring '07
 LEWIS,M.
 Markov chain, Xn, discrete time Markov, recurrent states, Mark E. Lewis, P10 P00 P01

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