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Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX]\leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$?

While it might appear intuitive that positively scaling a random variable should not change the sign of its mean, the possible correlation between N and X might complicate things. So far, I have had no luck proving it. On the other hand, I could not also come up with a counterexample. That is, I could not think of any RV $X$ (with positive mean) which upon multiplying with another positive random variable $N$ with bounded support, makes the mean of the product $NX$ negative.

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Of course not. E.g., suppose that $P(X=-1)=P(X=2)=1/2$. Suppose also that (for some real $n\ge1$) $N=n$ when $X=-1$ and $N=1$ when $X=2$. Then $EX>0$, whereas $ENX=-\frac12\,n+\frac12\,2<0$ if e.g. $n=3$.

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