I think that this is false, but I cannot think of a counterexample. Can anyone help?
EDIT: I am interested in the case when X,Y are not independent.
2026-03-28 13:40:25.1774705225
If $E[XY]=0$, then does that imply $E[X]=0$ or $E[Y]=0$?
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Suppose you are tossing a fair coin. Let $X=1$ if you get $H$ and $0$ if you get $T$. Let $Y=1$ if you get $T$ and $0$ if you get $H$.
To be sure, if $X,Y$ are independent variables then $E[XY]=E[X]\times E[Y]$. See, e.g., this. That's enough to show what you want under the, strong, assumption of independence.