Must the mean of a product of two random variable with positive mean positive?

Question

I have a form like X(X+Y-1) for X and Y independent random variables. I believe I can show $E[X+Y-1]>0$ and $E[X]>0$. Am I allowed to conclude that also $E[X(X+Y-1)]>0$? I have heard a fkg inequality, but don't know if it can be used in this multiple variables case Thank you in advance for any hints!

Answer 1

$$E[X(X+Y-1)] = E[X^2] + E[XY] -E[X] \geq E[X]^2 +E[X]E[Y] -E[X] = E[X](E[X] + E[Y] - 1) = E[X] E[X+Y-1] > 0 $$