Probability of positive inner product with expectation of one variable

Question

Assume that $\mathbb{P}(X\cdot Y > 0) > 0$ and $\mathbb{E}|X| < \infty$. Can we say that $\mathbb{P}(\mathbb{E}[X]\cdot Y > 0) > 0$?

Edit: Also assume that $\mathbb{P}(X\cdot Y \ge 0) = 1$.

Answer 1

As Robert points out, $X=Y$ and $E[X]=0$ gives a counterexample. If $E[X]\neq 0$, here is another counterexample:

Let $(X,Y)$ take values $(-1,-2), (0,-1), (2,0)$, each with probability $1/3.$

Note $$E[X]=1/3,P(XY>0)=P(X=-1,Y=-2)=1/3, P(XY\geq 0)=1$$ but $$P(E[X]Y>0)=P(Y>0)=0.$$

Answer 2

Consider $X$, $Y$ such that $\mathbb{P}(X=-5) = \frac{1}{2}$, $\mathbb{P}(X=1)=\frac{1}{2}$, $Y>0$.

Then $\frac{1}{2} = \mathbb{P}(X\cdot Y>0) > 0$, $\mathbb{E}[X] = -2 <\infty$ but $\mathbb{P}\left(\mathbb{E}[X]\cdot Y> 0\right) =\mathbb{P}\left(Y<0\right) =0 $.

Edit: In case your condition is $\mathbb{E}|X| <\infty $ instead of $\mathbb{E}[X] <\infty $ (I don't know it that's a typo) the example still stands.