Let $Y_i = \beta x_i + e_i $, where $e_1 ~ N(0, \sigma^2)$ and $e_2 ~ N(0, 2\sigma^2)$, and $e_1$ and $e_2$ are statistically independent. If $x_1 = 1$ and $x_2 = -1$ obtain the weighted least squares estimate of $\beta$ and find the variance in your estimate.
In the textbook which the problem is given the answer is $\beta^* = 1/3(2Y_1-Y_2)$ and $Var[\beta^*] = 2/3\sigma^2$. I do not have the solution but I get the same answers when I use GLS which is the page in the textbook right above that problem.
My question is why do they state that $e_1$ and $e_2$ are statistically independent ? I thought GLS is used the errors are correlated or are those two different things ? What am I missing ? Please help, it is something fundamental.
They are using weighted least squares, which is the special case of GLS where the errors are uncorrelated.
Weighted least squares can be used to deal with errors that have different variances (i.e. the usual assumption of homoskedasticity is not satisfied), which is the case in your example.