The expected value of the product of two random variables is equal to the product of the expected value, assuming that the variables are independent.
Statement: If the two variables X and Y are independent we have that the expectation of XY is equal to the product of the expectation of X and the expectation of Y.
E(XY) = E(X) E(Y) assuming that X and Y are independent.
Proof: We have the formula for covariance given as,
Cov(X,Y) = E(XY) – E(X) E(Y).
Since X and Y are independent we have that Cov(X, Y) = 0.
Substituting Cov(X, Y) = 0 in the above formula we get that,
E(XY) = E(X) E(Y).
