Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 27K / Solution

A random row vector has a multivariate normal distribution when every real linear combination of its coordinates is normally distributed, allowing a zero-variance constant. Its law is determined by its mean and covariance. For the conditional and testing formulas assume $\Sigma$ is positive definite, so both coordinate variances and the conditional variance are positive; the given ratio already requires ${\sigma }_{XX}>0$.

Put $Z=Y-\beta X$, with $\beta ={\sigma }_{XY}/{\sigma }_{XX}$. The pair $(X,Z)$ is jointly normal and $\mathrm{Cov}(X,Z)={\sigma }_{XY}-\beta {\sigma }_{XX}=0$, so $Z$ is independent of $X$. Its mean is ${\mu }_Y-\beta {\mu }_X$, and its variance is ${\sigma }_{YY}-{\sigma }_{XY}^2/{\sigma }_{XX}$. HenceStandardize each coordinate with its own positive standard deviation. The standardized pairs have mean zero and covariance $\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)$. Subtracting means and multiplying coordinates by positive constants leaves the sample correlation unchanged. Therefore the distribution of $r$ depends only on $\rho$.

For $n>2$, the stated statistic reduces algebraically toThis is strictly increasing from $(-1,1)$ onto $\mathbb{R}$, with derivative $\sqrt{n-2}(1-r^2{)}^{-3/2}$. Under $\rho =0$, change variables in the given Student density: $1+t^2/(n-2)=1/(1-r^2)$. HenceThe normalizing integral is $B(1/2,(n-2)/2)$. To test $\rho =0$ against a two-sided alternative at level $\alpha$, compute $t(r)$ and reject if $|t(r)|>t_{n-2,1-\alpha /2}$, equivalently ifA one-sided alternative uses the corresponding one-sided tail. Singular covariance gives $|r|=1$ almost surely when both variances are nonzero, and is outside the nondegenerate density calculation.