2. Two independent measurements, X and Y , are taken of a quantity . Assume that E(X) = E(Y ) = , but X = sd(X) and Y = sd(Y ) are unequal. The two measurements are combined by means of a weighted average to give Z = X + (1 )Y where is a scalar weight and [0, 1]. (a) Show that E(Z) = for any [0, 1]. (b) Show that V ar(Z) = 2 2 X + (1 w) 2 2 Y (c) Find in terms of X and Y that minimizes V ar(Z). (d) Under what circumstances is it better to use the average (X +Y )/2 than either X or Y alone?