Let the following classical linear regression:
$$y_i = x_i \theta + u_i, \quad E(u_i|x_i) \sim N(0, \sigma^2)$$
Can I conclude that $x$ and $u$ are independent?
I would like this because I want to prove that: $y_i|x_i \sim N(\theta x , \sigma^2)$. And I need the independence between $x_i$ and $u_i$.to use the linearity of the variance:
$$V(y_i|x_i) = V(x_i \theta|x_i) + V(u_i|x_i) $$