Differentiate LDA and QDA; compute boundary

For class 0, the mean is $\mu_0 = (0,0)$ and the covariance is $\Sigma_0 = \begin{bmatrix}1&0\\0&1\end{bmatrix}$. For class 1, the mean is $\mu_1 = (2,0)$ and the covariance is $\Sigma_1 = \begin{bmatrix}4&0\\0&1\end{bmatrix}$. The prior probabilities are $\pi_0=0.7$ and $\pi_1=0.3$. (1) State the LDA discriminant and decision rule when you (incorrectly) assume $\Sigma_0=\Sigma_1$, and provide the linear boundary equation. (2) State the QDA discriminant using the actual covariances and derive the quadratic boundary as an explicit scalar equation in $x$ and $y$. (3) Classify the point $x=(1,1)$ under both LDA and QDA, showing the numerical values of the discriminants. (4) Discuss when QDA is preferable and when LDA is safer with $n=60$ per class, and propose a regularized QDA that shrinks each class covariance toward a shared $\Sigma$ using a tunable shrinkage parameter; explain how you would select it via cross-validation.

Overview: This question tests understanding of Gaussian generative classification and discriminant analysis: how equal versus class-specific covariances affect linear (LDA) and quadratic (QDA) decision boundaries, how to compare discriminants numerically, and how to select models with regularization and cross-validation.