Statistics questions - help!!?

Suppose S is an unbiased estimator of 胃 if E[S] = 胃, otherwise it was biased, with bias = E[S] 鈥?胃. Find an unbiased estimator of the area (A) of a square plate with sides of length L. Suppose that we make two independent measurements L(1), L(2) of the length of the sides of the plate. Suppose that L (I) is Normally distributed with mean L and variance 蟽^2. Let your first estimator of A be the square of the average length of the two measurements and let your second estimator be the average of the squares of the first measurement and the second measurement. That is,


S(1) = [0.5(L(1)+ L(2))]^2 and S(2) = [0.5(L^2(1) + L^2(2))]. Which of these estimators has the least bias? If both are biased, can you think of a third estimator that is unbiased?





Well, I know that if X and Y are two independent random variables then E[XY] = E[X]E[Y], and that VAR[X] = E[X2]-[E[X]]^2





Any ideas?





Thanks!|||Alright.





E(S1) = E{(L1/2 + L2/2)^2}


= Var(L1/2 + L2/2) + E^2(L1/2+L2/2) (Uses the fact that Var(X) = E(X^2) - E^2(X))


= 0.25(sigma^2) + 0.25(sigma^2) + (L/2 + L/2)^2 (Since L1 and L2 are independent)


= 0.5(sigma^2) + L^2.





So the bias of S1 is 0.5(sigma^2)





E(S2) = E{0.5(L1^2 + L2^2)}


= 0.5E(L1^2) + 0.5E(L2^2)


= 0.5(Var(L1) + E^2(L1)) + 0.5(Var(L2) + E^2(L2))


= 0.5(sigma^2) + 0.5L^2 + 0.5(sigma^2) + 0.5L^2


= sigma^2 + L^2





So the bias of S2 is sigma^2.





So that means that S1 has the lower bias.





Perhaps S3 = L1L2 is unbiased?





E(L1L2) = E(L1)E(L2) (Since both are independent)


= L*L


= L^2





Bias is zero. There you go!|||Pal, you have a lot of guts voting for yourself. Jerk.

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|||i hope i don't take the same classes as u