Let $X$ follow a Normal distribution, and let $Y = a+bX$ (not both constants zero). The normal distribution is closed under shifting and scaling, so $Y$ also follows a normal distribution.
If I recall correclty, there exists "Cramer's condition" for bivariate normality, where we get it if all linear combinations of two random variables have a univariate normal distribution. In our case it is easy to see that any $Z = \delta_1X + \delta_2 Y$ will follow a Normal distribution too (not both constants zero). So it appears that $(X,Y)$ are jointly normal, with correlation coefficient equal to unity, in absolute value.
The question is the following: the joint support of $(X,Y)$ is a line in $\mathbb R^2$, and as a $(n-1)-$dimensional object in a $n-$dimensional space, it has Lebesgue measure zero. This I think, implies that the joint probability density function does not exist.
Q1: Is the above correct?
Q2: What are the main practical consequences of not being allowed to define the joint density (like, indicatively, what kind of statistical work we cannot do, and what kind we still can do, what statistical methods are no longer applicable, and which are still applicable)?
I guess the situation generalizes to any distribution that is closed under affine transformations, and when we want to assume in addition that the bivariate vector follows the 2-D analogue of the marginal distributions.