Let X and Y be random variables. The covariance Cov(X, Y) is defined by (see Exercise 6.2.23)

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Let X and Y be random variables. The covariance Cov(X, Y) is defined by (see Exercise 6.2.23) cov (X, Y) = E((X − μ(X))(Y − μ(Y))) . (a) Show that cov(X, Y) = E (XY) − E(X) E(Y). (b) Using (a), show that cov(X, Y) = 0, if X and Y are independent. (Caution: the converse is not always true.) (c) Show that V (X + Y ) = V (X) + V (Y ) + 2cov(X, Y ).
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