(i) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ on $\{1,2,3,4,5,6,7\}$ with generator matrix

$$Q=\left(\begin{array}{rrrrrrr} -6 & 2 & 0 & 0 & 0 & 4 & 0 \\ 2 & -3 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & -5 & 1 & 2 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 2 & 0 & -6 & 0 & 2 \\ 1 & 2 & 0 & 0 & 0 & -3 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & -2 \end{array}\right)$$

Compute the probability, starting from state 3 , that $X_{t}$ hits state 2 eventually.

Deduce that

$$\lim _{t \rightarrow \infty} \mathbb{P}\left(X_{t}=2 \mid X_{0}=3\right)=\frac{4}{15}$$

[Justification of standard arguments is not expected.]

(ii) A colony of cells contains immature and mature cells. Each immature cell, after an exponential time of parameter 2, becomes a mature cell. Each mature cell, after an exponential time of parameter 3, divides into two immature cells. Suppose we begin with one immature cell and let $n(t)$ denote the expected number of immature cells at time $t$. Show that

$$n(t)=\left(4 e^{t}+3 e^{-6 t}\right) / 7$$