Under what circumstances is it better to use the average (X+Y)/2 than either X or Y alone?
Here is what I have: (which may or may not be correct!)
We know E(X) = E(Y)= μ
Var((X+Y)/2) = (E((X+Y)/2)2) - (E((X+Y)/2))2
I found that E((X+Y)/2)= 0.5 E(X+Y) = 0.5[E(X) + E(Y)] = 0.5(2μ) = μ
and that (E((X+Y)/2)2)= .25[E(X2+2XY+Y2)]= .25[E(X2)+E(2XY)+E(Y2)]= 0.25[E(X2)+2E(X)E(Y)+E(Y2)]= 0.25[E(X2)+2μ2 +E(Y2)]
But now I am stuck! Am I on the right track, and what can I do now?