3. Consider the simple linear regression model $Y_i = \beta_0 + \beta_1 X_i + \epsilon_i$ with normal error, so $\epsilon_i$ are iid $N(0, \sigma^2)$ random variables. Given $n$ data points ${(X_i, Y_i)}_{i=1}^n$, let $b_0$ and $b_1$ be the least square estimates for $\beta_0$ and $\beta_1$. The $i$-th residual, $e_i$, is $e_i = Y_i - \hat{Y}_i = Y_i - b_0 - b_1 X_i$.
(a) For $i = 1$, derive the distribution for $e_1$. Hint: Write $e_1$ as $e_1 = \sum_j c_j Y_j$ for some choice of $c_1, c_2, \dots, c_n$.
(b) Are the residuals independent? Hint: what do we know about the relationship between $e_i$'s?
