# Are two Random Variables Independent if their support has a dependency?

This might be a really dumb question, but in a joint PDF of $$X$$ and $$Y$$, $$f_{XY}(x,y)$$, if the support of a random variable $$Y$$ depends on $$X$$, are the two random variables necessarily dependent? For example, if one has $$f_{XY}(x,y)=1/x$$, where $$0, then can one tell that $$X$$ and $$Y$$ are not independent simply by examining the support and not even looking at the $$1/X$$? My intuition suggests they can't be independent.

• An additional thought: I'm starting to think the they can't be independent since we can incorporate the support as indicator variables in the PDF. Is this correct? – StatsStudent May 9 '17 at 1:53
• $y$ shows up in the support -- the entire point of my question. – StatsStudent May 9 '17 at 2:11
• @SmallChess The support of $Y$ is wherever $Y$ has non-zero density. It's standard mathematical terminology. See en.wikipedia.org/wiki/Support_(mathematics)#Formulation. – Clarinetist May 9 '17 at 2:45
• Support is quite standard, indeed. The terminology is used in nearly every statistics textbook I've ever come across. The terminology is the right term and is used in statistics. – StatsStudent May 9 '17 at 3:16

I've determined that if there is a dependency between $X$ and $Y$ in the support of a bivariate pdf, then $X$ and $Y$ cannot be independent. To be sure, there is a Lemma (4.2.7 in Casella and Berger's Statistical Inference, 2d) that states: Let ($X$,$Y$) be a bivariate random vector with joint pdf or pmf $f(x,y)$. Then $X$ and $Y$ are independent random variables if and only if there exists functions $g(x)$ and $h(y)$ such that for every $x$ $\in\mathbb{R}$ and $y$ $\in\mathbb{R}$:
$f(x,y)=g(x)h(y)$
If we incorporate the support (e.g. $0<y<x<1$) as an indicator function in the joint pdf (e.g. ${f_{XY}(x,y)=xy}I_{(0<y<x<1)}$ then the joint PDF cannot be written as a product of only $g(x)$ and only $h(y)$, so $X$ and $Y$ cannot be independent.)