Moran processes - exercises

1. Describe the transition probabilities for the Moran process with neutral drift.
2. Obtain the transition probability matrix for the Moran process with neutral drift with $N=4$ individuals.
3. State and prove the theorem for fixation probabilities in a birth death process.
4. Extend the formulae of question 3 to the case of a Moran process on a game.
5. For the following games, obtain the fixation probability $x_1$ for $N=4$:
   1. $A=\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$
   2. $A=\begin{pmatrix}1 & 2 \\ 3 & 1\end{pmatrix}$
6. Consider the game $A=\begin{pmatrix}r & 1 \\ 1 & 1\end{pmatrix}$ for $r>1$ and $N$, and obtain $x_1$ as a function of $r$. How does $r$ effect the chance of fixation?