# In Simple Linear Regression $\hat \beta_1$ and $\bar Y$ are independent [duplicate]

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?