What is the meaning of the $0<x, y<\infty$ bound in this joint PDF?

Question

I'm given a joint PDF,

\begin{equation} f_{X,Y}(x,y) = \begin{cases} e^{-(\frac{x}{y}+y)}y^{-1} & \text{ , } 0<x,y<\infty\\ 0 & \text{ , otherwise} \end{cases}. \end{equation}

From which my professor asks us to find the marginal PDF $f_{Y}(y)$. He says: "Clearly, $f_{Y}(y)=0$ for $y\leq 0$." I do not understand why $f_{Y}(y)=0$, this is not clear to me. As far as I can see, if $y\leq0$, then $y<\infty$ and therefore $f_{Y}(y)\neq0$ for $y\leq 0$. So what am I not seeing?

Answer 1

If $0<x,y<\infty$ is equivalent to

If $0<x$ and $0<y$ and $x<\infty$ and $y<\infty$.

For $y\le0$, $y$ is indeed strictly less than $\infty$, but $0<y$ does not hold. Thus the condition is not satisfied and, by definition, we have $f_{X,Y}(x,y)=0$.