I have 2 questions about continuous-time Markov Chain.
Let's consider the following example.
Be $\{X(t),t\geq 0\}$ a continuous time Markov chain with state space $S_X=\{1, 2, 3\}$ and the following transition matrix \begin{equation*} Q = \begin{pmatrix} -4& 2 & 2\\1.5 & -2.5 & 1 \\ 0&1&-1\end{pmatrix}\end{equation*} Finally, we have $X(0)=3$.
I would like to know how to compute 2 differents probabilities.
First, the probability that the chain is in the state 1 just after the second jump. About this one, I've noticed that the only path from 3 to 1 is to go from 3 to 2 to 1. However, I don't have any idea about how to compute this.
Also, the probability that the chain stays in state 3 until time t=1. I have no idea how to do this. Cause you give me some hint ?
Thanking you all in advance !
N.B. : I never had any courses about continuous time markov chain, and I found the books about this subject not helpful, so sorry if my questions are a little dumb.