(a) Define the Moran model and Kingman's $n$-coalescent. State and prove a theorem which describes the relationship between them. [You may use without proof a construction of the Moran model for all $-\infty<t<\infty$.]

(b) Let $\theta>0$. Suppose that a population of $N \geqslant 2$ individuals evolves according to the rules of the Moran model. Assume also that each individual in the population undergoes a mutation at constant rate $u=\theta /(N-1)$. Each time a mutation occurs, we assume that the allelic type of the corresponding individual changes to an entirely new type, never seen before in the population. Let $p(\theta)$ be the homozygosity probability, i.e., the probability that two individuals sampled without replacement from the population have the same genetic type. Give an expression for $p(\theta)$.

(c) Let $q(\theta)$ denote the probability that a sample of size $n$ consists of one allelic type (monomorphic population). Show that $q(\theta)=\mathbb{E}\left(\exp \left\{-(\theta / 2) L_{n}\right\}\right)$, where $L_{n}$ denotes the sum of all the branch lengths in the genealogical tree of the sample - that is, $L_{n}=\sum_{i=2}^{n} i\left(\tau_{i}-\tau_{i-1}\right)$, where $\tau_{i}$ is the first time that the genealogical tree of the sample has $i$ lineages. Deduce that

$$q(\theta)=\frac{(n-1) !}{\prod_{i=1}^{n-1}(\theta+i)}$$