Problem 1 The random variable X is normally distributed. We wish to test that X is over-dispersed. Denote by Y the indicator variable that X has variance larger than 1: (X]Y=0)~N'(0,1) (X|Y=1)~N(0,01)where01>1 Denote by p0 = IP{Y = 0} the prior probability that X is distributed as a standard normal. 1. Assume that the loss incurred from a false negative is k: times larger than the loss incurred from a false positive. Find the decision rule that minimizes the expected loss. 2. Assume Y(:c) = 1 {|:c| 2 1'}. Express the FPR and TPR as a function of T. Sketch the ROC curve for various values of 01 > 1. (Hint: the scipy. stats .norm in Python might come in handy) 3. Find the optimal decision rule that balances the expected number of false positives and false nega- tives. Find the tradeoff factor is: from part 1 that corresponds to this decision rule for p0 = 1/ 3, 01 = 2. (Hint: you can use numerical solvers to get an approximate solution)