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Assume $X$ and $Y$ are two positive RVs and $Cov(X,Y)>0$. Does this imply that $Cov(X,1/Y)<0$, or is more information needed?

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No, this is not implied. The sign of a covariance is essentially only preserved in a consistent way by linear transformations: for all other functions, including $f(x) = x^{-1}$ you can exploit the curvature of the function to make the sign whatever you want.

Here's a quick example I got by playing around with the numbers: suppose you sample $(1,1)$, $(2.5, 0.1)$ and $(3,2)$ uniformly to generate $(X,Y)$ pairs. This gives positive covariance, and still does if we replace the $Y$ values by $1, 10, 0.5$.

There may be numerically simpler examples available, but at least three points are necessary.

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