Data $(x_1, y_1), \dots, (x_n, y_n)$ is modelled with $x_i$ being non random and $y_i$ being observed values of $$Y_i = \alpha + \beta (x_i - \bar x) + \sigma \epsilon_i$$ with $\epsilon_i \sim N(0,1)$.
I have calculated the MLE's of the parameters $\alpha, \beta$ and $\sigma^2$ with $\hat \alpha = \bar y$ and $$\hat \beta = \frac{\sum y_i(x_i - \bar x)}{\sum (x_i - \bar x)^2}$$ we now have a corresponding estimator for $\beta$ given by $$B = \frac{\sum Y_i(x_i - \bar x)}{\sum (x_i - \bar x)^2}$$ which apparently (according to my notes) is normally distributed with a mean of $\beta$ and a variance of $$\sigma^2 \over \sum(x_i - \bar x)^2$$ and I can't justify to myself why this is the case! Could someone please help explain this?
self-study
tag. You might like to read its tag wiki info. $\endgroup$