Given K different classes (i.e.,



!1,!2,


,!K

), we assume each class !k ¹k = 1, 2,


, Kº is modeled by a multivariate Gaussian distribution with the mean vector k and the covariance matrix ; that is, p¹x j !k º = N¹x j k , º, where  is the common covariance matrix for all K classes. Suppose we have collected N data samples from these K classes (i.e., fx1, x2,


, xN g), and let fl1, l2,


, lN g be their labels so that ln = k means that the data sample xn comes from the kth class !k . Based on the given data set, derive the MLE for all model parameters (i.e., all mean vectors k ¹k = 1, 2,


, Kº) and the common covariance matrix .