Define the Moran model. Describe briefly the infinite sites model of mutations.

We henceforth consider a population with $N$ individuals evolving according to the rules of the Moran model. In addition we assume:

• the allelic type of any individual at any time lies in a given countable state space $S$;

• individuals are subject to mutations at constant rate $u=\theta / N$, independently of the population dynamics;

• each time a mutation occurs, if the allelic type of the individual was $x \in S$, it changes to $y \in S$ with probability $P(x, y)$, where $P(x, y)$ is a given Markovian transition matrix on $S$ that is symmetric:

$$P(x, y)=P(y, x) \quad(x, y \in S)$$

(i) Show that, if two individuals are sampled at random from the population at some time $t$, then the time to their most recent common ancestor has an exponential distribution, with a parameter that you should specify.

(ii) Let $\Delta+1$ be the total number of mutations that accumulate on the two branches separating these individuals from their most recent common ancestor. Show that $\Delta+1$ is a geometric random variable, and specify its probability parameter $p$.

(iii) The first individual is observed to be of type $x \in S$. Explain why the probability that the second individual is also of type $x$ is

$$\mathbb{P}\left(X_{\Delta}=x \mid X_{0}=x\right),$$

where $\left(X_{n}, n \geqslant 0\right)$ is a Markov chain on $S$ with transition matrix $P$ and is independent of $\Delta$.