What is Var(X - Y)? We know that the formula for

Alan Smith 2022-01-14 Answered
What is Var(X - Y)?
We know that the formula for Var(X + Y) is Var(X) + Var(Y) + 2Cov(X,Y)
Does this mean that for Var(X - Y) it is just:
Var(X) - Var(Y) - 2Cov(X,Y)?
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Expert Answer

enhebrevz
Answered 2022-01-15 Author has 25 answers
Var(X)=E[X2]E[X]2
The definition of variance.
Var(XY)=E[(XY)2]E[XY]2E[X22XY+Y2]E[XY]2
Linearity of expectation:
E[X22XY+Y2]=E[X2]+E[Y2]2E[XY]andE[XY]=E[X]E[Y]
Var(XY)=E[X2]2E[XY]+E[Y2](E[X]22E[X]E[Y]+E[Y]2)E[X2]E[X]2+E[Y2]E[Y]22(E[XY]E[X]E[Y])
Now note that:
Cov(x,y)=E[(xE[x])](yE[y])]
=E[xy]E[xE[y]]E[y[E[x]]+E[x]E[y]
=E[xy]E[x]E[y]E[y]E[x]+E[x]E[y]
=E[xy]E[x]E[y]
Which immediately gives the result desired in terms of the covariance.
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Marcus Herman
Answered 2022-01-16 Author has 41 answers
It will be Var(X)+Var(Y)−2Cov(X,Y), because Var(−Y)=Var(Y).
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