# How to prove independence based on condition for linear regression

Suppose that we have the following linear model $$\mathbb{E}[Y_{i}]=b_{0}+b_{1}(x_{i}-\bar{x})$$ where $$i=1,...,n$$ and $$\bar{x}$$ is the mean of $$x_{i}$$ and $$Y_{i}$$ are uncorrelated with constant variance $$\sigma^{2}$$. We derived that the least squares estimators of $$b_{0}$$ and $$b_{1}$$ are the following

$$\hat{b_{0}} = \bar{Y}$$ and $$\hat{b_{1}}=\frac{\sum_{i}(x_{i}-\bar{x})Y_{i}}{\sum_{i}(x_{i}-\bar{x})^{2}}$$

We also know from the covariance matrix of the estimators that $$\mathbb{E}[\hat{b_{0}}\hat{b_{1}}]=0$$ are uncorrelated.

Now we are given a different linear function $$\tilde{b}=\sum_{i}b_{i}Y_{i}$$ where $$b_{i}$$ are constants. And we are also given the mean square error of $$\tilde{b}$$ which is

$$MSE(\tilde{b}) = var(\tilde{b})+(\mathbb{E}(\tilde{b})-b_{0})^{2}$$.

My goal is to prove that $$MSE(\tilde{b})$$ is independent of $$b_{1}$$ if and only if $$\sum_{i}b_{i}(x_{i}-\bar{x})=0$$

Thus we have to prove that

i) $$\mathbb{E}[MSE(\tilde{b})b_{1}]=\mathbb{E}[MSE(\tilde{b})]\mathbb{E}[b_{1}]\Rightarrow \sum_{i}b_{i}(x_{i}-\bar{x})=0$$

ii) with $$\sum_{i}b_{i}(x_{i}-\bar{x})=0 \Rightarrow \mathbb{E}[MSE(\tilde{b})b_{1}]=\mathbb{E}[MSE(\tilde{b})]\mathbb{E}[b_{1}]$$

But I am struggling on how to prove it.