Probability | Part IA, 2004

Define the covariance, cov(X,Y)\operatorname{cov}(X, Y), of two random variables XX and YY.

Prove, or give a counterexample to, each of the following statements.

(a) For any random variables X,Y,ZX, Y, Z

cov(X+Y,Z)=cov(X,Z)+cov(Y,Z)\operatorname{cov}(X+Y, Z)=\operatorname{cov}(X, Z)+\operatorname{cov}(Y, Z)

(b) If XX and YY are identically distributed, not necessarily independent, random variables then

cov(X+Y,XY)=0\operatorname{cov}(X+Y, X-Y)=0

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