1. Assume, in addition. that X and Y are independent. Prove the following: E[X Y] = E[X] E[Y]
var(X + Y) = var(X)+var(Y)
Solution
Note that the definition of Covariance(X, y) = E(XY) - E(X)E(Y). for independent
random variables Covariance(X,Y) = 0 so for this case where X and Y are independent we have
Covariance(X, Y) = E(XY) - E(X)E(Y) 0 = E(XY) - E(X)E(Y) E(XY) = E(X)E(Y)
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