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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.