Under what conditions is E[XY]= E[X] E[Y] ? If X and Y re white noise then why would this equation hold true?
The equation will hold true if X and Y uncorrelated. Independence is not required. Two white noise processes are independent, which is more than enough for E[XY]= E[X]E[Y].
We know the distribution of two random variables is the product of marginal distribution if they are independent. More precisely, when their covariance is zero.
If X and Y are Gaussian white noise, the equation might or might not hold true because you can have two correlated white noise.