patter2809
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- Joined
- Mar 29, 2013
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- 17
Need to show Var(X+Y) = Var(X) + Var(Y) given that X,Y are independent RVs.
A. Var(X+Y) = E(X+Y)2 + E2(X+Y)
= E(X2+2XY+Y2) + E(X+Y)E(X+Y)
= E(X2) + 2E(XY) + E(Y2) + E2(X) + 2E(X)E(Y) + E2(Y)
= [E(X2) + E2(X)] + [E(Y2) + E2(Y)] + 2[E(XY) + E(X)E(Y)]
= Var(X) + Var(Y) + 2[E(XY) + E(X)E(Y)]
Is it wrong to say E(X2+2XY+Y2) = E(X2) + 2E(XY) + E(Y2) given that X2 and 2XY are not independent? Or have I gone wrong in some other way?
Thank you for your time.
A. Var(X+Y) = E(X+Y)2 + E2(X+Y)
= E(X2+2XY+Y2) + E(X+Y)E(X+Y)
= E(X2) + 2E(XY) + E(Y2) + E2(X) + 2E(X)E(Y) + E2(Y)
= [E(X2) + E2(X)] + [E(Y2) + E2(Y)] + 2[E(XY) + E(X)E(Y)]
= Var(X) + Var(Y) + 2[E(XY) + E(X)E(Y)]
Is it wrong to say E(X2+2XY+Y2) = E(X2) + 2E(XY) + E(Y2) given that X2 and 2XY are not independent? Or have I gone wrong in some other way?
Thank you for your time.