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

If X and Y are independent random variables with equal variances, find Cov(X+Y,X−Y).
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Raven Mosley
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, then
Cov(X,Y)=0
Next, remember that
Cov(X,X)=Var(X).
Therefore, due to bilinearity and symmetry properties of the covariance, we have that
Cov(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 that
Cov(X,Y)=Cov(Y,X)
because by definition
Cov(X,Y)=E[(X-E(X))*(Y-E(Y))],
which is clearly the same as
Cov(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's 0).
Result:
Cov(X+Y,X-Y)=0

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