= 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 $\operatorname{Cov}(X,Z)=\sigma_{XY}-\beta\sigma_{XX}=0$, so $Z$ is independent of $X$. Its mean is $\mu_Y-\beta\mu_X$, and its <variance> is $\sigma_{YY}-\sigma_{XY}^2/\sigma_{XX}$. Hence
$$
\boxed{Y\mid X=x\sim N\left(\mu_Y+\beta(x-\mu_X),\ \sigma_{YY}-\frac{\sigma_{XY}^2}{\sigma_{XX}}\right).}
$$
Standardize each coordinate with its own positive <standard deviation>. The standardized pairs have mean zero and <covariance> $\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}$. Subtracting means and multiplying coordinates by positive constants leaves the sample correlation unchanged. Therefore \b[the distribution of $r$ depends only on $\rho$].
For $n>2$, the stated statistic reduces algebraically to
$$
\boxed{t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}.}
$$
This 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: $1+t^2/(n-2)=1/(1-r^2)$. Hence
$$
\boxed{p_r(r)=\frac{\Gamma((n-1)/2)}{\sqrt\pi\,\Gamma((n-2)/2)}(1-r^2)^{(n-4)/2},\qquad-1<r<1.}
$$
The 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 if
$$
|r|>\frac{t_{n-2,1-\alpha/2}}{\sqrt{n-2+t_{n-2,1-\alpha/2}^2}}.
$$
A 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.
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