# Thread: Sampling Distribution of the Estimated Regression Mean

1. ## Sampling Distribution of the Estimated Regression Mean

I have finally just proved the sampling distribution of the estimated regresion mean $\displaystyle \hat{\alpha} + \hat{\beta}x_{0}$

as

$\displaystyle \hat{\alpha} + \hat{\beta}x_{0} \sim N (\alpha + \beta x_{0} , \sigma^{2} \{ \frac{1}{n} + \frac{(x_{0}-\bar{x})^2}{S_{xx}} \})$
After all that I now have been asked to comment on the parameters and estimated values about which are fixed quantities and which are random variables, but from all my work proving the formula I have got myself completely confused on this commenting section.

2. ## Re: Sampling Distribution of the Estimated Regression Mean

Hey klw289.

Random variables have a distribution and fixed quantities are constants.

Estimators are random variables, so the question you need to ask is a) what are you estimating and b) what is the estimator for those parameters and what is its PDF?

Hint: In this model you have a simple linear regression with two parameters. This should start you off in the right direction.

3. ## Re: Sampling Distribution of the Estimated Regression Mean

Thanks for the hint. Its now all sorted. A big thank you