TITLE: Markov Chain Expected value
QUESTION [5 upvotes]: Let $(X_n)_{n\in\mathbb{N}}$ be a Markov chain with State space $E=\{1,2,3\}$ and transission matrix 
$$P=\begin{bmatrix} 0 & 1/3 & 2/3 \\        1/4 & 3/4 & 0 \\ 2/5 & 0 & 3/5     \end{bmatrix}$$
How can I compute $E[X_{n+1}^3|X_n=j]$, for $j=1,2,3$.

REPLY [5 votes]: Can't you just calculate it? For example,
$E[X_{n+1}^3 | X_n = 1] $ 
$= 1^3.Pr(X_{n+1} = 1 | X_n = 1) +  2^3.Pr(X_{n+1} = 2 | X_n = 1) +  3^3.Pr(X_{n+1} = 3 | X_n = 1)$ 
$ = 1 \cdot 0 + 8 \cdot (1/3) + 27 \cdot (2/3) $
$ = ...$