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Suppose that $X$ and $Y$ are positive and square-integrable random variables, such that $X$ and $Y$ are positively correlated, i.e., $\mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] \geq 0$. Let $f: \mathbb{R} \to \mathbb{R}_+$ be a positive and increasing function. My question is: Are $f(X)$ and $f(Y)$ positively correlated? Thanks very much.

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Let $X\sim U(0,2)$, $Y=(X-1)^2$, and $f(x)=(\sqrt{x}+1)1\{x\ge 0\}+e^x1\{x<0\}$. Then $f$ is a positive, increasing function and $\mathrm{Cov}(X,Y)=0$. However, $$ \mathrm{Cov}(f(X),f(Y))=\left(\frac{4}{15}-\frac{\sqrt{2}}{5}\right)<0. $$

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  • $\begingroup$ Thanks a lot for your example! $\endgroup$ – Richie Jun 3 '17 at 6:07

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