Problem 2 (10pt): [Likelihood Ratio] Suppose we consider two category classification, the class conditionals are assumed to be Gaussian, i.e., $p(x|\omega_1) = N(2, 1)$ and $p(x|\omega_2) = N(6, 1)$, based on prior knowledge, we have $P(\omega_2) = \frac{1}{3}$. We do not penalize for correct classification, while for misclassification, we put 1 unit penalty for misclassifying $\omega_1$ to $\omega_2$ and put 2 units for misclassifying $\omega_2$ to $\omega_1$. Derive the bayesian decision rule using likelihood ratio.
