Suppose $X,Y$ are two positively correlated Gaussians with zero mean and unit variance. Is it the case that for $a,b \in \mathbb{R}$, $$ \Pr[X \leq a, Y \leq b] \geq \Pr[X \leq a] \Pr[Y \leq b]? $$
2026-03-26 01:03:35.1774487015
CDF of positively correlated Gaussians
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It may be overkill, but I believe this follows from the fact that for Gaussians, positive (resp. negative) association is equivalent to positive (resp. negative) correlation [1], along with the definition of positive association [2] (apply it to $f = \mathbf{1}_{(-\infty,a]}$ and $g= \mathbf{1}_{(-\infty,b]}$).
[1] Pitt, Loren D. Positively Correlated Normal Variables are Associated. Ann. Probab. 10 (1982), no. 2, 496—499. doi:10.1214/aop/1176993872. https://projecteuclid.org/euclid.aop/1176993872
[2] Dubhashi, Devdatt, and Desh Ranjan. Balls and bins: A study in negative dependence. Random Structures & Algorithms 13.2 (1998): 99—124. https://www.brics.dk/RS/96/25/BRICS-RS-96-25.pdf