# Expected value of markov chain URGENT!

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• October 20th 2009, 10:23 PM
mtlchris
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!
• October 20th 2009, 11:19 PM
CaptainBlack
Quote:

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!

The distribution over states after three steps where $A$ is the single step transition matrix and $x_0$ the initial distribution over states is:

$x_3=x_0A^3=[1/2, 1/2, 0]A^3$

From this distribution you calculate the expectation for the state.

CB