Consider the following model:

\(X_1,…,X_n \stackrel{iid}{\sim} f(x), \quad Y_i = \theta X_i + \varepsilon_i, \quad \varepsilon_i \stackrel{iid}{\sim} \mbox{N}(0,\sigma^2).\)

1. Compute \({\mathbf E }(Y \mid X)\)
2. Compute \({\mathbf E }(\varepsilon \mid X)\)
3. Compute \({\mathbf E }( \varepsilon)\)
4. Show \( \theta = \frac{{\mathbf E}(XY)}{{\mathbf E}(X^2)}\)