What choice of and minimizes the variances,subject to the condition of unbiasedn

Question

What choice of and minimizes the variances,subject to the condition of unbiasedness? Hint: The problem doesn't say that the surveys were based onsampling without replacement. So despite the notation, do notassume that either of two estimates is the sample mean. Please help me!! Need why above are the answer. What choice of alpha and beta minimizes the variances,subject to the condition of unbiasedness? Hint: The problem doesn't say that the surveys were based onsampling without replacement. So despite the notation, do notassume that either of two estimates is the sample mean. Answer should be Two surveys were independently conducted to estimate apopulation mean, ?. Denote the estimates and their standarderrors by X1 and X2 are unbiased. For some alpha and beta, the twoestimates can be combined to give a better estimator: X1 and X2 , and sigma x1 and sigma x2 . Assume that

Explanation / Answer

i think to be unbiased, alpha and beta must add to 1, so thatthe expected value of X remains equal to the mean mu, no matterwhat X1 nd X2 are.
i'm not sure if i've a rigorous proof as to why the values arewhat they are, but my intuition is that you want to scale alpha andbeta according to their standard errors. that is, the one with thebigger error gets the smaller scale factor. since there are onlytwo terms in X=alpha*X_1 + beta*X_2, then it should make sense toassign alpha to be the error of X_2 and beta to be the error of X_1(both alpha and beta are then normalized to ensure alpha+beta =1)...
not sure if i'm completely useless here...

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