# Randomly scaled random variables

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.

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. $$n=3$$.