Suppose that are independently and identically distributed variables in a random sample drawn from a population with mean and variance .
a) Determine the mean and variance of .
b) Construct your own unbiased estimator for the population mean, and calculate its variance.
I'm not entirely sure what I'm doing for a) and I don't know what to do for b) at all. Can anyone help?
I think I track the hidden sense of your problem. They expect you to compare different population mean estimators including sample mean that looks like (X_1+...X_n)/n because it has the smallest variance among unbiased pop mean estimators (Gauss-Markov) provided X_i have identical variances. You can choose any combination g_1*X_1+...+g_n*X_n with g_1+...+g_n=1 as unbiased pop mean estimate. If they have the same means but different variances the optimal weights g_i are proportional to
1/(sigma_i)^2. Let's say e.g. g_i=1/(sigma_i)^2 / ( 1/(sigma_1)^2+...+1/(sigma_n)^2 )