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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 $Y$?

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  • $\begingroup$ 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 $1$ $\endgroup$ – Henry Sep 17 '20 at 11:09
  • $\begingroup$ 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. $\endgroup$ – whuber Sep 17 '20 at 14:35

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