Consider a continuous time Markov chain with state space {1, 2, 3} and transition rates $$q(1,2) = 2, q(1, 3) = 3$$ $$q(2, 1) = 0, q(2, 3) = 5 $$ $$ q(3, 1) = 1, q(3, 2) = 0$$ Calculate $P(X_{7.5} = 2 | X_0 = 1)$.
We haven't yet discussed these types of conditional probabilities in class, and our textbook doesn't do a great job of explaining the method in which we should calculate them. All I can find at the moment is the following lemma: $$P(X_{s+t} = j | X_s = i) = \frac{\pi(j) p_{t}(j, i)}{\pi(i)}$$ Thus, in this case, the answer would simply be $$P(X_{7.5} = 2 | X_0 = 1) = \frac{\pi(2) p_{7.5}(2, 1)}{\pi(1)}$$ I can easily find this stationary distribution, but I'm not sure how to think about $p_{7.5}(2,1)$. I believe we can find an imbedded discrete time markov chain for this process, but I'm still not sure how we would find $p_{7.5}(2, 1)$.
Any push in the right direction is much appreciated, thank you!