# Best Choice of Estimator. How to compute Variance of Estimator.Basis for it?

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.

It's possible to derive the Expectation of $$X_i^2$$, however there's a much simpler way to calculate your variance:
$$Var(X1) = Var(\frac{1}{2n} \sum_{i=1}^{2n} X_i) \\ = \frac{1}{4n^2} \sum_{i=1}^{2n} Var(X_i) \\ = \frac{1}{4n^2} 2 n \sigma^2 \\ = \frac{\sigma^2}{2n}$$ This comes from the fact that the variance is linear.