Is a bivariate normal distribution is only possible (for rv $X$ and $Y$) if $aX+bY \forall a,b \in R$ is normal?

Question

On this Wikipedia page it is stated that:

To say that the pair $(X,Y)$ of random variables has a bivariate normal distribution means that every linear combination $aX+bY$ of $X$ and $Y$ for constant (i.e. not random) coefficients $a$ and $b$ has a univariate normal distribution.

Is this to say that if $aX+bY$ does not have an univariate distribution for any $a$ and $b$ then the joint density $f_{XY}$ is not bivariate normal? If so, is there a way to show this?

Answer 1

Yes, a finte collection of random variables are jointly normal iff any linear combination of them is normal. However degenerate random variables are to be considered normal for this to be true.