Let $X$ and $Y$ be two non-negative random variables and be negative correlated, i.e.,
$$\mathbb{E}[XY] \leq \mathbb{E}[X]\mathbb{E}[Y].$$
Let $h(\cdot)$ and $g(\cdot)$ be two non-negative, monotone increasing functions. Do we have
$h(X)$ and $g(Y)$ being negative correlated as well? Intuitively this quite makes sense to me..
2026-03-25 16:06:17.1774454777
Does negative correlation survive monotone transformation?
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Copying from Rahul's comment since it answers the question:
No. Counterexample: Consider the discrete uniform distribution on {(0,1),(0.9,0),(1,1)}, and let h be the monotone function that maps {0,0.9,1} to {0,0.1,1}.