I am working on an exercise asking which of the two following estimators; $X1, X2 $for the population mean of a normal population with parameters $ \mu, \sigma$ is best and why
$X1:=\frac {X_1+X_2+....X_{2n}}{2n}$,
$X2:=\frac {X_1+X_2+....X_n}{n}$
I computed the expectation and both are unbiased estimators. I tried to find the variance of the estimators, but the computations became unwieldy:
$Var(X1)= E[(X1- \mu)^2]=$
$\frac {1}{4n^2}E[( \Sigma _{i=1}^{2n}X_i-\mu)^2]=$
$E[\frac{1}{4n^2} (X_1+X_2+...+ X_{2n})^2 - 2\mu X1+ \mu^2]$
and in order to clear this I need to compute $E[X_i^2] $ ,
Which I don't see how to do. I tried standardizing but did not see it helping. Without this computation, I cant decide the best estimator for the pop mean based on neither variance nor bias-variance tradeoff.
Any ideas?
Thanks.