Expectation of Product of Random Variables Proof From the definition of the expected value, the expected value of the product of two random variables is E(X . Y) = EErr-r2 . P(X=Y =r2) where the sum is over all possible values of r1 and r2 that the variable X and Y can take on_ Using the definition above formally prove that if the events X = r1 and Y =r2 are independent (for any r1 and r2) , we have E(X . Y) = E(X) . E(Y). Now; if you have two random variables X and Y and Y=aI+c, show that if E[X2] # E[XJ? , then E(X . Y) # E(X) . E(Y). If you roll a die 6 times in a rOW what is the expected value of the product of the 6 outcomes?