I throw two dice and record the scores $S_{1}$ and $S_{2}$. Let $X$ be the sum $S_{1}+S_{2}$ and $Y$ the difference $S_{1}-S_{2}$.

(a) Suppose that the dice are fair, so the values $1, \ldots, 6$ are equally likely. Calculate the mean and variance of both $X$ and $Y$. Find all the values of $x$ and $y$ at which the probabilities $\mathbb{P}(X=x), \mathbb{P}(Y=y)$ are each either greatest or least. Determine whether the random variables $X$ and $Y$ are independent.

(b) Now suppose that the dice are unfair, and that they give the values $1, \ldots, 6$ with probabilities $p_{1}, \ldots, p_{6}$ and $q_{1}, \ldots, q_{6}$, respectively. Write down the values of $\mathbb{P}(X=$ 2), $\mathbb{P}(X=7)$ and $\mathbb{P}(X=12)$. By comparing $\mathbb{P}(X=7)$ with $\sqrt{\mathbb{P}(X=2) \mathbb{P}(X=12)}$ and applying the arithmetic-mean-geometric-mean inequality, or otherwise, show that the probabilities $\mathbb{P}(X=2), \mathbb{P}(X=3), \ldots, \mathbb{P}(X=12)$ cannot all be equal.