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?