# Do two dependent variables have a functional relationship?

Suppose $$X$$ and $$Y$$ are two dependent random variables (e.g. they are the elements of a bivariate normal distribution with $$\rho\ne0$$). Is it true that there always exists a function $$f$$ such that $$f(X,Y)=0$$?

If the statement is not generally true, are there any special cases? Moreover, how the $$f$$ in question (if any) could be found, given the pdfs of $$X$$ and $$Y$$?

• No not always (excluding the trivial example) unless $\rho =\pm 1$, in which case there will be a function where $f(X,Y)=0$ with probability $1$ – Henry Sep 17 '20 at 11:09
• Trivially, yes: let $f$ be constantly zero. If $f$ is nontrivial and differentiable almost everywhere, then Calculus teaches us that the set where $f(X,Y)=0$ is a union of one-dimensional manifolds: that is, the distribution of $(X,Y)$ is degenerate. For bivariate Normal variables this occurs only when $\rho=\pm 1.$ You don't need to analyze the PDFs or know much at all about the bivariate Normal distribution to draw these conclusions, which are general. – whuber Sep 17 '20 at 14:35