Consider a gaussian linear model with random regressors
$X_i \sim N(\mu_X,\sigma_X^2), 1 \leq i \leq n$, i.i.d.
$Y_i | X_i \sim N(\beta_0+\beta_1 X_i,\sigma_\varepsilon^2)$
Are either of the ordinary least squares coefficient estimates $\hat\beta_0$ or $\hat\beta_1$ correlated with the regressors $X_i$?
My intuition says no, because the mean of the coefficient estimates depends only the true coefficients. Increasing $X_i$ increases the mean of $Y_i$ such that the two effects somehow cancel in computing the OLS estimates $\hat\beta$.