Definition of Support I am a bit confused with the definition regarding support. Say the joint density function exists. For the random variables $X$ and $Y$, is the joint support of $X$ and $Y$:
the values $X$ and $Y$ can take st. $f_{X,Y}(x,y) >0$?
Or is the support of $f_{X,Y}(x,y)$ the values $X$ and $Y$ can take st. $f_{X,Y}(x,y) >0$
In other words is it defined in terms of Random variables or probability density functions? Or are they both correct?

Support of random variable $X$ is a set of all values, that the random variable has. So if $S_X$ is support of $X$ and $S_Y$ is support of random variable $Y$ and they are independent then the support of random vector $(X,Y)$ is Cartesian product $S_X \times S_Y$.
If the vector $(X,Y)$ has density $f_{X,Y},$ the the support is set
$$S_{X,Y} = \left\{(x,y) \in S_X \times S_Y: f_{X,Y}(x,y) >0 \right\}.$$ 
