Two independent measurements, X and Y , are taken of a quantity μ. E(X) = E(Y ) = μ, but σX and σY are unequal. The two measurements are combined by means of a weighted average to give
Z = αX + (1 − α)Y
where α is a scalar and 0 ≤ α ≤ 1.
a. Show that E(Z) = μ.
b. Find α in terms of σX and σY to minimize Var(Z).
c. Under what circumstances is it better to use the average (X + Y )/2 than either X or Y alone?
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