Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 11F (Probability)

For every $u\in {\mathbb{R}}^2$,The right-hand side is a normal random variable by the definition of a bivariate normal distribution. Therefore every linear combination of the coordinates of $AX$ is normal, so $\boxed{AX\text{ is bivariate normal}}$.

For every $\lambda \in {\mathbb{R}}^2$, the scalar ${\lambda }^{T}X$ is normal with mean ${\lambda }^T\mu$ and variance ${\lambda }^{T}V\lambda$. The moment-generating function of a normal distribution therefore givesThus $\mu$ and $V$ determine the moment-generating function on all of ${\mathbb{R}}^2$. By the uniqueness theorem for moment-generating functions, they uniquely determine the probability distribution of $X$.

The covariance matrix isIf $\mathrm{Cov}(X_1,X_2)=0$, the moment-generating function in part (ii) factors into the product of the two marginal moment-generating functions. Thus $X_1$ and $X_2$ are independent random variables; this is the independence of uncorrelated jointly normal variables.

By bilinearity of covariance,ChooseThen $(X_1,Y)$ is bivariate normal by part (i) and has zero covariance, so $Y$ is a normal random variable independent of $X_1$. Hence $X_2=a{X}_1+Y$.

Let $D=R-L$ be right-arm length minus left-arm length. A linear combination of jointly normal random variables is normal, andwhileThus $D\sim N(2,3)$, and standardization of a normal random variable gives