If X and Y are independent random variables with equal variances, find Cov(X+Y,X−Y).

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Raven Mosley · Accepted Answer

We are given that X and Y are independent random variables such that Var(X)=Var(Y), and we need to evaluate Cov(X+Y,X−Y).First, since X and Y are independent, thenCov(X,Y)=0Next, remember thatCov(X,X)=Var(X).Therefore, due to bilinearity and symmetry properties of the covariance, we have thatCov(X+Y,X-Y)=Cov(X,X)-Cov(X,Y)+Cov(Y,X)-Cov(Y,Y)=Var(X)-Cov(X,Y)+Cov(X,Y)-Var(Y)=Var(X)-Var(Y)=0.The symmetry property of the covariance was not mentioned before, but it is obvious thatCov(X,Y)=Cov(Y,X)because by definitionCov(X,Y)=E[(X-E(X))*(Y-E(Y))],which is clearly the same asCov(Y,X)=E[(Y-E(Y))*(X-E(X))].Also, notice that this result still holds even without the assumption of independence of X and Y, since the value Cov(X,Y) gets canceled anyway (without assuming it&#039;s 0).Result:Cov(X+Y,X-Y)=0

If X and Y are independent random variables with equal variances, find Cov(X+Y,X−Y).

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