1. Suppose that we have p-dimensional data X generated from two normal distributions labelled Y = 1 and Y = 2, with means µ1, µ2 and common covariance matrix E. Assume the prior probabilities are each 0.5. Derive an expression for the posterior probabilities P(Y = j|X = x). Relate your answer to logistic regression.2. You are given some data by a collaborator, and asked to build a two-class classifier with n = 1000 observations and p = 500 features, to predict the risk of a customer defaulting on a loan. Unfortunately about 25% of the features are missing at random (and not the same 25% each time). The result is that nearly every observation has some missing features. How would you deal with this?3. In the same setting as the previous question, you later learn that some of the features like monthly income are not missing at random, but are more likely to be missing because the mortgage company has lost track of the customer. How would you deal with this issue?
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