Recall that a random variable $X$ in $\mathbb{R}^{2}$ is bivariate normal or Gaussian if $u^{T} X$ is normal for all $u \in \mathbb{R}^{2}$. Let $X=\left(\begin{array}{c}X_{1} \\ X_{2}\end{array}\right)$ be bivariate normal.

(a) (i) Show that if $A$ is a $2 \times 2$ real matrix then $A X$ is bivariate normal.

(ii) Let $\mu=\mathbb{E}(X)$ and $V=\operatorname{Var}(X)=\mathbb{E}\left[(X-\mu)(X-\mu)^{T}\right]$. Find the moment generating function $M_{X}(\lambda)=\mathbb{E}\left(e^{\lambda^{T}} X\right)$ of $X$ and deduce that the distribution of a bivariate normal random variable $X$ is uniquely determined by $\mu$ and $V$.

(iii) Let $\mu_{i}=\mathbb{E}\left(X_{i}\right)$ and $\sigma_{i}^{2}=\operatorname{Var}\left(X_{i}\right)$ for $i=1,2$. Let $\rho=\frac{\operatorname{Cov}\left(X_{1}, X_{2}\right)}{\sigma_{1} \sigma_{2}}$ be the correlation of $X_{1}$ and $X_{2}$. Write down $V$ in terms of some or all of $\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}$ and $\rho$. If $\operatorname{Cov}\left(X_{1}, X_{2}\right)=0$, why must $X_{1}$ and $X_{2}$ be independent?

For each $a \in \mathbb{R}$, find $\operatorname{Cov}\left(X_{1}, X_{2}-a X_{1}\right)$. Hence show that $X_{2}=a X_{1}+Y$ for some normal random variable $Y$ in $\mathbb{R}$ that is independent of $X_{1}$ and some $a \in \mathbb{R}$ that should be specified.

(b) A certain species of East Anglian goblin has left arm of mean length $100 \mathrm{~cm}$ with standard deviation $1 \mathrm{~cm}$, and right arm of mean length $102 \mathrm{~cm}$ with standard deviation $2 \mathrm{~cm}$. The correlation of left- and right-arm-length of a goblin is $\frac{1}{2}$. You may assume that the distribution of left- and right-arm-lengths can be modelled by a bivariate normal distribution. What is the probability that a randomly selected goblin has longer right arm than left arm?

[You may give your answer in terms of the distribution function $\Phi$ of a $N(0,1)$ random variable $Z$. That is, $\Phi(t)=\mathbb{P}(Z \leqslant t)$.J