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Let's say fitted regression line is $y_i=b_1+b_2x_i$.That is, $b_1$ and $b_2$ are estimators.

The text book says that if $e_i$ (residual) is normally distributed, then $b_1$ and $b_2$ are also normally distributed.

The evidence given in the text book is that if the normality of $e_i$ implies the normality of $y_i$. Then, since $b_2=\sum w_iy_i$, $b_2$ is the sum of normal distribution, which is normal distribution.

But, there is no explanation with regard to $b_1$. The evidence given is $b_1=\overline y - \overline x b_2$. Can we prove that $b_1$ is also normally distributed from this?? or is there another way to show it?

Thank you in advance.

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You can use the same reasoning as for $b_2$:

  1. $\bar{y}$ is Normally distributed since it's a sum of Normal distributions
  2. $b_2$ is Normally distributed as you suggest
  3. $\bar{x}$ is a number (i.e. a constant, your observations), which just scales $b_2$

So $b_1$ is the difference between two Normal distributions, which as you know, is a Normal distribution.

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