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

Question

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

V a r (X) = E [X^{2}] - E {[X]}^{2}

The definition of variance.

V a r (X - Y) = E [{(X - Y)}^{2}] - E {[X - Y]}^{2} E [X^{2} - 2 X Y + Y^{2}] - E {[X - Y]}^{2}

Linearity of expectation:

E [X^{2} - 2 X Y + Y^{2}] = E [X^{2}] + E [Y^{2}] - 2 E [X Y] and E [X - Y] = E [X] - E [Y]

V a r (X - Y) = E [X^{2}] - 2 E [X Y] + E [Y^{2}] - (E {[X]}^{2} - 2 E [X] E [Y] + E {[Y]}^{2}) E [X^{2}] - E {[X]}^{2} + E [Y^{2}] - E {[Y]}^{2} - 2 (E [X Y] - E [X] E [Y])

Now note that:

C o v (x, y) = E [(x - E [x])] (y - E [y])]

= E [x y] - E [x E [y]] - E [y [E [x]] + E [x] E [y]

= E [x y] - E [x] E [y] - E [y] E [x] + E [x] E [y]

= E [x y] - 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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