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Under what conditions is E[XY]= E[X] E[Y] ? If X and Y re white noise then why would this equation hold true?

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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].

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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.

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If X and Y are Gaussian white noise, the equation might or might not hold true because you can have two correlated white noise.

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    $\begingroup$ "covariance zero" is less precise than "independent" (since independent pairs of variables will have covariance 0 but the converse doesn't hold). It's the more general situation that's needed. $\endgroup$ – Glen_b Mar 30 '16 at 12:41

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