This is an econometrics exercise in which we were asked to show some properties of the estimators for the model $$Y=\beta_0+\beta_1X+U$$ where we were told to assume that $X$ and $U$ are independent.
This exercise has many parts, in one of its parts I have shown that $$\sqrt{n}(\hat{\beta_1}-\beta_1) \sim N\bigg(0, \frac{\sigma^2}{Var(X)}\bigg) $$
$$\implies \hat{\beta_1} \sim N \bigg(\beta_1, \frac{\sigma^2}{n Var(X)} \bigg)$$
where $n$ is the sample size of $X$, and $\sigma^2$ is the variance of $U$. The last questions asks
Why is the assumption that $X$ and $U$ are independent important for you answer in the distribution above?
I don't really know how to answer this. I am tempted to say that independence actually is not the minimum assumption that is necessary, so we can actually have some weaker assumption (maybe something like mean independence or $Cov(X, U)=0$) and that distribution would still be true....
I am not very confident in my answer and I hope someone can help me.
Note that the OLS of $\beta_1$ is $$ \hat{\beta}_1= \frac{ \sum(x_i - \bar{x})y_i }{ \sum(x_i - \bar{x})^2 }. $$ Its expectation and variance derived under the assumptions that $ \mathbb{E}[\epsilon|X] = 0 $ and $ \mathbb{V}[\epsilon|X] = \sigma^2 , $ namely, that both these quantities are independent of $X$. In fact, you may conclude it using only the assumption of uncorrelated $X$ and $\epsilon$.