compare the expected values $E[X(Y-1)]$ and $E[X]E[Y-1]$ while $0 < Y < 1$

Question

X and Y are two positive variables with unknown distributions. It's known that 0 < 1 < Y, and Cov(X,Y) > 0.

Cov(X,Y) is an observation based on real data.

Question - given which circumstance can we find that

E[X(Y-1)] > E[X]E[Y-1] so that Cov(X,Y) > 0?