Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 28J / b

Put $s=\lambda +\mu$. Solving either Kolmogorov equation givesThese entries have the correct initial values and their rows sum to one.

Direct multiplication gives $Q^2=-sQ$, hence for $n\ge 1$,Therefore the matrix exponential iswhich expands to exactly the transition matrix found in part (i).

Writing $\pi =({\pi }_1,{\pi }_2)$, the equations $\pi Q=0$ and ${\pi }_1+{\pi }_2=1$ giveThis is the invariant distribution. Since $e^{-st}\to 0$, both rows of $P(t)$ converge to $\pi$, directly verifying convergence to equilibrium.

The Markov property and conditioning on the intermediate state give the bridge probabilitySubstitution from part (i) yields