So here's the question:
Suppose we observe Xi (as i=1,..., 10) which are identically distributed with mean E(Xi) = mu and variance V(Xi)=sigma^2. Then is x(bar) = X1 an unbiased estimator of mu?
Logically speaking, it doesn't make sense that it would be. If i=1,..., 10 is the entire population, then the mean is 55/10 (sum of Xi as i goes to 10, divided by 5) = 5.5. To say that an estimator for the mean, x(bar), is equal to X1 (which if i=1), in turn, equals 1. Is this unbiased? It doesn't seem so - population mean is 5.5 and that is not equal to 1. Am I understanding this correctly?