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For any three real random variables $X,Y,Z$, prove or disprove that $\langle XY\rangle$, $\langle XZ\rangle$, $\langle YZ\rangle$ can't all be negative.

Here $\langle \cdot \rangle$ denotes an expectation over the joint probability distribution of the three variables.

Intuitively I think they can't all be negative, because if $X,Y$ are anticorrelated, and $X,Z$ are anticorrelated, how can $Y,Z$ also be anticorrelated? But I am not sure how to prove this, or if there is a counterexample.

It might be necessary to assume that $X,Y,Z$ have zero means.

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marked as duplicate by Glen_b Sep 24 '18 at 14:18

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  • $\begingroup$ You easily can have three variables all with negative pairwise correlation, though there's a limit on how negative you can make them. $\endgroup$ – Glen_b Sep 24 '18 at 13:47
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    $\begingroup$ (a) If the inner product between two vectors is negative, this is equivalent to saying what about the angle $\theta$ between the vectors? Hint1: $\cos \theta_{xy} = \frac{\langle x, y \rangle}{\|x\| \|y\|}$ (b) Can you draw a picture where every angle $\theta_{xy}, \theta_{yz}, \theta_{xz}$ satisfies the conditions in (a)? $\endgroup$ – Matthew Gunn Sep 24 '18 at 14:21
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    $\begingroup$ Hint2: $\langle x, y \rangle > 0$ implies $\cos \theta_{xy} > 0$ hence $\theta_{xy}$ is an acute angle. What does $\langle x, y \rangle < 0$ imply? $\endgroup$ – Matthew Gunn Sep 24 '18 at 14:22
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    $\begingroup$ @Becko Can you draw three vectors that have obtuse angles between them? That point away from each other? $\endgroup$ – Matthew Gunn Sep 24 '18 at 14:30
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    $\begingroup$ To generate a trivariate sample with population correlations all -1/2, try this (in R): n <- 10000; u <- rnorm(n,0,0.5); v <- rnorm(n,0,sqrt(0.75)); xyz <- data.frame(x=u+u,y=-u+v,z=-u-v); cor(xyz) $\endgroup$ – Glen_b Sep 24 '18 at 16:17