I know that if X and Y are independent variables then the expectation value of X and Y i.e $ E(XY)=E(X).E(Y).$ But then the expectation value will be equal to zero only when either of them i.e $E(X) or E(Y) $ is zero. Also, I have read somewhere that each term averages out to zero. But this can happen only for certain datasets/distributions right? How to generalize this?
This concept is frequently used while deriving standard deviation of sampling standard deviations, also in statistical mechanics where we drop cross terms averages.
