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Suppose $f$ and $g$ are monotonically increasing, and bounded, and let $X$ be a random variable. I want to show $f$, $g$ have positive covariance. I tried to compute it directly but I am not getting anything useful

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"It is generally taken for granted that the covariance of two increasing functions of a random variable is positive. The present paper contains an elementary proof of this fact."

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The proof presented in this paper is ingenious because it avoids introducing an independent copy of $X$. But the author vastly overstates his case asserting that The inequality holds indeed, but a proof is difficult to find in the literature; an exception is the book by Schürger [1998; Aufgabe 4.22] who suggests a proof based on the independent product of two probability spaces. In fact, the proof is arch classical, see for example the very first pages of Hermann Thorisson's paper Coupling Methods in Probability Theory in the Scandinavian Journal of Statistics dating back from 1995. – Did Oct 21 '11 at 22:11

I think you can consider $\mathbb{E}[(f(X)-f(Y))(g(X)-g(Y))]$ where $X$ and $Y$ are i.i.d.

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