Expected value of markov chain URGENT!

i have a 3 state (0,1,2) markov chain with transition matrix p=
1/2, 1/3, 1/6
0, 1/3, 2/3
1/2, 0, 1/2
and that p(x0= 0) = p(x0= 1)= 1/2

and i need to fond E[X3]

so i find p^3 by matrix multiplication to be:
13/36, 11/54,47/108
4/9, 4/27, 11/27
5/12, 2/9, 13/36

now the solution says the answer is 1/2(47/108)+1/2(11/54)+1/4(13/36)

but i have no idea how this is obtained! please help!

Quote:

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Originally Posted by mtlchris

i have a 3 state (0,1,2) markov chain with transition matrix p=
1/2, 1/3, 1/6
0, 1/3, 2/3
1/2, 0, 1/2
and that p(x0= 0) = p(x0= 1)= 1/2

and i need to fond E[X3]

so i find p^3 by matrix multiplication to be:
13/36, 11/54,47/108
4/9, 4/27, 11/27
5/12, 2/9, 13/36

now the solution says the answer is 1/2(47/108)+1/2(11/54)+1/4(13/36)

but i have no idea how this is obtained! please help!

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The distribution over states after three steps where is the single step transition matrix and the initial distribution over states is:



From this distribution you calculate the expectation for the state.

CB