Classification problem

Question

I have this question in the assignment and I just don't know how to proceed.

Answer 1

Here, you are asked to find the most likely class of $x$, or in other words, to perform Maximum Likelihood Estimation.
For any $i\in\{1,2,3\}$, by applying Bayes' theorem, we have that $$\begin{align}\mathbb P(Y=i\mid X=x) &=\frac{f_{X\mid Y=i}(x) \times \mathbb P(Y=i)}{f_X(x)} \\ &= \frac{f_{X\mid Y=i}(x) }{3\times f_X(x)}\\ &\propto f_{X\mid Y=i}(x)\end{align}$$ We do not need to explicitly compute the terms whose value are independent of the value of $Y$.
Now, we know that conditional on $Y = i$, $X$ follows a multivariate normal distribution with mean $\mu_i$ and covariance matrix $\Sigma_i$. The conditional density of $X\mid Y=i$ is thus $$f_{X\mid Y=i}(x) = \frac{\exp\left(-\frac{1}{2}(x-\mu_i)^T\Sigma_i^{-1}(x-\mu)\right)}{\sqrt{(2\pi)^2|\Sigma_i|}} $$ To classify $x$, you have to pick the class $i$ that maximizes $\mathbb P(Y=i\mid X=x) $.
We see that it is equivalent to choosing the class $i$ that maximizes $f_{X\mid Y=i}(x) $, so now you have all the elements you need to classify $x$.