Continuous marginal distributions do not imply continuous joint distribution

Question

I already proved the other implication. I need to find an explicit example that shows that if there is some random vector $(X,Y)$ and $X$ and $Y$ have both continuous marginal distributions, then distribution of ($X,Y$) is not necessarily continuous type.

Any help please?

Answer 1

Hint: Consider $(X,Y) := (X,X)$ for a continuous random variable $X$. Show that $$D := \{(x,y) \in \mathbb{R}^2; x=y\}$$

has Lebesgue measure zero, but $\mathbb{P}((X,Y) \in D)=1$. Conclude that $(X,Y)$ is not absolutely continuous (with respect to the Lebesgue measure).

Answer 2

Take $X$ and $Y$ as being fully correlated bivariate normal random variables. Then the joint density is given by dirac masses as follows: $$f(x, y) = g(x)\delta_x(y)$$ where $g(x)$ is a standard normal density.