How do you prove 2 normal random variables X and Y are jointly normally distributed?

Question

How do you prove 2 normal random variables X and Y are jointly normally distributed? I know that any linear sum of X and Y should be normally distributed but how do you prove that?

Answer 1

If you know $X$ is normally distributed and $Y$ is normally distributed, and nothing else is known, then it is not possible to prove that the pair $(X, Y)$ are jointly normally distributed, due to the existence of pathological examples.

One simple way to demonstrate that $X$ and $Y$ are jointly normal is to find another pair $(U,V)$ of independent normal variables such that $X$ and $Y$ are each a linear combination of $(U,V)$, i.e., $$\begin{bmatrix}X\\Y\end{bmatrix} =\begin{bmatrix}a&b\\ c&d\end{bmatrix} \begin{bmatrix}U\\V\end{bmatrix}+ \begin{bmatrix}e\\f\end{bmatrix} $$ for some constants $a,b,c,d,e,f$. Depending on the author, this is either a definition or a result that requires proof. If it requires proof, a common approach is to use moment-generating functions.