I was given the following exercise:
Let $(X,Y)$ be a random vector with density $$\frac{2}{y}\centerdot 1_{D}$$ where $D=\{ (x,y)\in (0,1)\times (0,\infty ) : 0<x<y, 0<xy<1\}$.
Determine whether $X$ and $Y$ have density.
Find the probability density function of $X$.
The only current method I know for, say, testing whether $X$ has a density function is to -assuming it has one- compute it and then check whether the found function is in fact its density function, similarly as done in a recent post of mine. However the way the problem is phrased seems to imply that one can determine whether $Y$ has a density function without having to compute it first. How can this be done?