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!