Use the formula Cov(X, Y) =E[XY]-E[X]E[Y] to prove the following properties of covari- ance for any...

 

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Use the formula Cov(X, Y) =E[XY]-E[X]E[Y] to prove the following properties of covari- ance for any random variables, X, X1, X2, Y, Y, and Y2. (a) Cov(X,X) = Var(X) = (b) Cov(X, Y) Cov(Y, X) (c) Cov(X1 X2,Y) = Cov(X, Y) + Cov(X2,Y) (d) Cov(X, Y+ Y2) = Cov(X, Y) + Cov(X, Y2). (Hint: This last property follows from (b) and (c)) Use the formula Cov(X, Y) =E[XY]-E[X]E[Y] to prove the following properties of covari- ance for any random variables, X, X1, X2, Y, Y, and Y2. (a) Cov(X,X) = Var(X) = (b) Cov(X, Y) Cov(Y, X) (c) Cov(X1 X2,Y) = Cov(X, Y) + Cov(X2,Y) (d) Cov(X, Y+ Y2) = Cov(X, Y) + Cov(X, Y2). (Hint: This last property follows from (b) and (c))