comparing the variance of estimators

Question

when we know that both estimators are unbiased, we prefer the one with a smaller variance.

so considering that we want to estimate the population mean using 5 data points, we consider the 2 estimators below:

a. Mu1 = (x1+x2+x3+x4+x5)/5 b. Mu2 = (x1+x2+x3)/3 + x4 - x5

var(Mu1) = σ²/5 var(Mu2) = (7/3)σ²

hence in comparing both estimators, estimator (a) would be better. however I need help in understanding the math behind getting the variance of both estimators!

Answer 1

Assuming independence, we have $$Var(X \pm Y) = Var(X)+Var(Y)$$

and also $$Var(kX)=k^2Var(X).$$

Hence,

$$Var\left( \frac{\sum_{i=1}^5X_i}5\right)=\frac1{25}\sum_{i=1}^5Var(X_i)=\frac{\sigma^2}5$$

$$Var\left(\frac{\sum_{i=1}^3X_i}{3} +X_4-X_5\right)=\frac19\sum_{i=1}^3Var(X_i)+Var(X_4)+Var(X_5)=\left( \frac13+2\right)\sigma^2$$