Hi I'm trying to find the variance of the sample mean below. I'm confused about the 4th and 5th lines, particularly where does the j' come from and why has it been introduced? Thanks any help appreciated.
$\overline{x}=\frac{1}{3n}\sum_{i=1}^{3}\sum_{j=1}^{n}X_{ij}$
$var(\overline{x})=var(\frac{1}{3n}\sum_{i=1}^{3}\sum_{j=1}^{n}X_{ij})$
$=\frac{1}{9n^{2}}\sum_{i=1}^{3}var(\sum_{j=1}^{n}X_{ij})$
$=\frac{1}{3n^{2}}E[\sum_{j=1}^{n}(X_{ij}-\mu )]^{2}$
$=\frac{1}{3n^{2}}\sum_{j=1}^{n}\sum_{j'=1}^{n}E[(X_{ij}-\mu)(X_{ij'}-\mu)]$
$=\frac{1}{3n^{2}}\sum_{j=1}^{n}E[(X_{ij}-\mu)^{2}]+2\sum_{j=1}^{n}\sum_{j'=j+1}^{n}E[(X_{ij}-\mu)(X_{ij'}-\mu)]$