is it possible to find X and Y such that E[X] is positive, Y is positive and E[XY] is strictly negative?

Question

If $X$ is an integrable real random variables such that $E[X] \ge 0$ and $Y$ is a positive integrable random variable is it possible that E[XY]<0 ?

Answer 1

Yes: $$ X(t) = 2 \text{ for $0 \le t \le 1$ } \\ X(t) = -1 \text{ for $-1 \le t < 0$ } \\ X(t) = 0 \text{ elsewhere,} $$ while $$ Y(t) = \frac{1}{1+t^2} \text{ for $t < 0$ } \\ Y(t) = .00001 \frac{1}{1+t^2} \text{ for $t \ge 0$ } \\ $$