Expectation of square of random variable and their mean.
Solution 1:
Hint: You just need the following facts:
- $Var(cX)=c^2Var(X)$ for some constant $c$
- $Var(X)+(EX)^2=E(X^2)$
- $Var(\sum X_i)=\sum Var(X_i)+\sum_i\sum_{j\neq i}Cov(X_i,X_j)$
- For independent random variables $X,Y$, $Cov(X,Y)=0$
Try using these facts. Otherwise,
Complete Proof: $Var(\bar{X})=Var\left(\frac{1}{n}\sum_{i=1}^nX_i\right)=\frac{1}{n^2}Var(\sum_{i=1}^nX_i)=\frac{1}{n^2}\sum_{i=1}^nVar(X_i)=\frac{\sigma^2}{n}$