Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 23K (Applied Probability)

Assume $h(x)>0$, so the conditioning event has positive probability. For a continuous-time Markov chain, write ${\tau }_y={\int }_0^{T_A}{1}_{\{X_s=y\}}ds$. At each time $s<T_A$, the Markov property gives $\mathbb{P}(T_A<\infty \mid {\mathcal{F}}_s)=h(X_s)$. Using Tonelli theorem for the nonnegative occupation integral,Division by $h(x)$ yields $\boxed{{\mathbb{E}}_x[{\tau }_y\mid T_A<\infty ]=[h(y)/h(x)]{\mathbb{E}}_x{\tau }_y}$. The discrete-time proof replaces the integral by a sum. Infinite expectations are allowed in the nonnegative sense; if $h(y)=0$, the numerator is zero directly, avoiding an ambiguous product of zero and infinity.

In the neutral Moran model, a fixed population of $N$ individuals carries either allele $a$ or its alternative. Each individual reproduces at rate one and replaces a uniformly selected individual. If $i$ carry $a$, the two changing-state rates are $q_{i,i+1}=q_{i,i-1}=i(N-i)/N$. States zero and $N$ are absorbing. This time normalization is essential for the stated expected time.

The hitting probability $h_i$ of $N$ before zero satisfies $h_{i+1}-2h_i+h_{i-1}=0$, with $h_0=0,h_N=1$. Thus $\boxed{h_i=i/N}$. Let $G_{ij}$ be expected occupation time at $j$ before absorption. The Green function equation in its starting index isIt is linear on either side of $j$. Matching values and the difference jump at $j$ givesFor example the difference jump is $-N/[j(N-j)]$, so multiplying by $j(N-j)/N$ gives exactly $-1$.

Condition on fixation using part (i), with $A=\{N\}$ and the occupation stopped on absorption at either endpoint; on paths absorbed at zero, hitting $N$ is impossible. Equivalently apply the Markov-property occupation proof directly to this event. The conditional occupation time at $j$ is $(j/i)G_{ij}$. For $j\ge i$ it is one; for $j<i$ it is $[(N-i)/i]j/(N-j)$. Summing over all transient states yieldsHere $1\le i\le N$; at $i=N$ the fixation time is zero, while at $i=0$ conditioning is undefined.