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Expected value of product of random variables

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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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