Population Analysis

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a) since u^ = C1 X1 + C2 X2 + ... + Cn Xn
then the expected value of u^ is:
E(u^)= E( C1 X1 + C2 X2 + ... + Cn Xn )
= E(C1 X1) + E(C2 X2) + ... + E(Cn Xn )
= C1 E(X1) + C2 E(X2) + ... + Cn E(Xn )

also, because E(Xi) = u for all i = 1, 2, ... n
Then E(u^)= C1* u + C2*u + ... + Cn*u
= (C1 + C2 + ... + Cn) u
so for u^ to be unbiased, we need E(u^) = u, or
(C1 + C2 + ... + Cn) u = u
therefore, (C1 + C2 + ... + Cn) = 1 is the necessary and sufficient condition for u^ to be ...