# Independence between error and regressor

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:

$$V(y_i|x_i) = V(x_i \theta|x_i) + V(u_i|x_i)$$

Some idea?

NO , You cannot conclude that $$x_i$$ and $$u_i$$ are independent just by by looking at an equation.
However in a classical regression setting it is assumed that $$x_i$$ are known values (aka constants , ie , they are not random variables). And Yes , they are assumed to be independent of $$u_i$$.
P.S : Just to remind you again that they are assumed to be independent. In time series context/presence of lagged variables, the $$x_i$$'s may not be independent of $$u_i$$'s.