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I want to show that, in simple linear regression $\hat\beta_1 $ and $\bar Y$ are independent.

My attempt: I have calculated the $\mathcal Cov(\hat \beta_1,\bar Y)$ and it turns out to be $0$.I also notice that $\hat \beta_1$ and $\bar Y$ both are normally distributed(Simply because, they are linear combination of $Y_i's$ and each $Y_i $ is normally distributed.). But if we have two uncorrelated normal random variables,that does not imply that they are independent. So I don't know how to show that they are actually independent? Any help would be appreciated.Thanks in advance.

My intuition is that somehow i have to calculate the joint pdf of $\hat \beta_1$ and $\bar Y$ and then the joint pdf simply splits into two independent functions of Single variables.Can anyone help me find the joint pdf?

Soumyadip Sarkar
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    Is this a question from a course or textbook? If so, please add the `[self-study]` tag & read its [wiki](https://stats.stackexchange.com/tags/self-study/info). – gung - Reinstate Monica Sep 23 '19 at 15:42
  • Search for [normal uncorrelated independent](https://stats.stackexchange.com/search?q=normal+uncorrel*+independent+score%3A1). – whuber Sep 23 '19 at 15:45
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    @whuber how do i know that they are jointly normal? – Soumyadip Sarkar Sep 23 '19 at 15:48
  • @SoumyadipSarkar Do you understand what to do if you can prove joint normality? – Dave Sep 23 '19 at 15:49
  • You state you "noticed" the normality. The principal ways in which that would be observed would immediately, with no calculation, demonstrate joint normality, so I wonder whether you could show us how you determined each statistic is separately normal? – whuber Sep 23 '19 at 15:51
  • @Dave yes,i know what to do after showing joint normality. – Soumyadip Sarkar Sep 23 '19 at 15:52
  • @whuber i am saying they are marginally normal but that does not mean they are jointly normal. – Soumyadip Sarkar Sep 23 '19 at 15:54
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    @SoumyadipSarkar What he means is to show how you know about marginal normality without knowing joint normality. – Dave Sep 23 '19 at 16:37
  • @Dave I have edited the question and said that they are marginally normal because each of them are linear combination of $Y_i's$ and each $Y_i$ is normal. – Soumyadip Sarkar Sep 23 '19 at 16:40
  • By assumption the $Y_i$'s are *jointly* normal. That's implied by the usual "iid" statement for the errors. – whuber Sep 23 '19 at 17:08
  • @whuber so is it true that if $Y_i's$ are jointly normal then any two linear combination of those are also jointly normal? – Soumyadip Sarkar Sep 23 '19 at 17:11
  • Yes: that is explained in various threads here, but it's easy to see just by writing down the joint distribution either as a density or a characteristic function, because that reduces the issue to whether a quadratic function of linear functions is quadratic (and of course it is). – whuber Sep 23 '19 at 17:12
  • @whuber but i am unable to write down the joint distribution function. – Soumyadip Sarkar Sep 23 '19 at 17:16
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    See https://stats.stackexchange.com/search?q=linear+combination+normal for help. Here's one of the top hits: https://stats.stackexchange.com/a/22883/919. Here's the same problem, specifically for OLS regression: https://stats.stackexchange.com/questions/133312. – whuber Sep 23 '19 at 17:18
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/99034/discussion-between-soumyadip-sarkar-and-whuber). – Soumyadip Sarkar Sep 24 '19 at 09:13

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