Let $(X_n)_{n \in N_0}$ be a Markov chain with the following graph:
(a) Find its probability transition matrix.
(b) Find all stationary (invariant) distributions of the Markov chain.
(c) Let the initial distribution be $X_0 = (0.5, 0.5, 0)$. Calculate the distribution of $X_2$ and $P(X_0·X_2 = 1)$.
Here's my work:
(a) $P = \begin{bmatrix}0 & 0.5 & 0.5 \\0.5 & 0.5 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$
(b) Since $3$ is the only absorbing state, the stationary distribution has to be $(0, 0, 1).$
(c) The distribution of $X_2 = X_0 \cdot P^2 = (\dfrac{1}{4}, \dfrac{3}{6}, \dfrac{3}{8}).$
I'm not sure how to solve for $P(X_0 \cdot X_2 = 1)$. Is the question looking for $P(X_0 = 1, X_2 = 1)?$