Let X and Y be random variables with means µX and µY . The covariance of...

Let X and Y be random variables with means µX and µY . The covariance of X and Y is given by, Cov(X, Y ) = E[(X − µX)(Y − µY )]

a) Prove the following three equalities: Cov(X, Y ) = E[(X − µX)Y ] = E[X(Y − µY )] = E(XY ) − µXµY

b) Suppose that E(Y |X) = E(Y ). Show that Cov(X, Y ) = 0 (hint: use the law of interated expectations to show that E(XY ) = µXµY ). In this case, what is the correlation coefficient ρ between X and Y , equal to?

c) Suppose that Cov(X, Y ) = 0. Does this imply that X and Y are independent? Explain your reasoning.

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Expert Solution

milcah answered 8 months ago

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