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 (X))(Y (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)...

a CovXY EXY XEY EXY XY EXY XY E ...

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

Question:

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 ).