# Justifying the distribution for the maximum likelihood estimator in a linear regression example

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?

• As this is basically standard bookwork, please add the self-study tag. You might like to read its tag wiki info. – Glen_b Jun 17 '13 at 0:17

Step 1: General: Recognize that $$(x_i - \bar x)$$, and its square, and $$\frac{1}{\sum (x_i - \bar x)^2}$$ are constants, not random variables. Further note that $$\alpha$$ and $$\beta$$ and $$\sigma$$ are also constants. First write $$B$$ as a constant times an expression containing a random variable. Focus carefully on the part that's not just a constant and then deal with that constant after you have that part worked out.
Step 2: Expectation: Remember that the expectation of a sum is the sum of expectations. Notice that you can take the expectation inside the summation and then move $$(x_i - \bar x)$$ outside that expectation. Notice you have another expression for $$Y_i$$ that you can use. Look again at step 1 and split up the expectation and pull out constants in the appropriate way. What's left is trivial to find the expectation of.
Step 3: Variance: Since the $$Y$$s are independent, the variance of a sum is the sum of the variances. You should also know a fact about the variance of a constant times a random variable to use here (more than once).