(a) What is a $Q$-matrix? What is the relationship between the transition matrix $P(t)$ of a continuous time Markov process and its generator $Q$ ?

(b) A pond has three lily pads, labelled 1, 2, and 3. The pond is also the home of a frog that hops from pad to pad in a random fashion. The position of the frog is a continuous time Markov process on $\{1,2,3\}$ with generator

$$Q=\left(\begin{array}{ccc} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 1 & 0 & -1 \end{array}\right)$$

Sketch an arrow diagram corresponding to $Q$ and determine the communicating classes. Find the probability that the frog is on pad 2 in equilibrium. Find the probability that the frog is on pad 2 at time $t$ given that the frog is on pad 1 at time 0 .