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Expected value of product of random variables


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) \text{ 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)

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