Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 38 / 5 / i / Solution

The fitted continuous-time multi-state model allows only adjacent fatigue-state jumps. Its four transition intensities, in reciprocal years, are shown here:

In state $2$, the total exit intensity is ${\lambda }_2=q_{21}+q_{23}=0.4684+0.3835=0.8519$. By the holding time law for a continuous-time Markov chain, the time until the next jump has an exponential distribution with this rate. ThereforeThe printed interval for $q_{22}=-{\lambda }_2$ gives the interval $(0.7219,1.005)$ for ${\lambda }_2$. Taking reciprocals reverses the endpoints, so the transformed $95\%$ confidence interval for the mean holding time isFor the next jump's destination, competing outgoing intensities giveIndeed the density of a first jump to $1$ at time $t$ is $q_{21}{e}^{-{\lambda }_{2}t}$; integrating gives the ratio above. This is a conditional next-jump probability, not a transition probability of being mild at a fixed later visit.