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) where 01 > 1 Denote by p0 = lP{Y = O} 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 f'(:c) = 1 {|x| 2 7'}. Express the FPR and TPR as a function of '7'. 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 negar tives. Find the tradeo\" factor 16 from part 1 that corresponds to this decision rule for 100 = 1/3, 01 = 2. (Hint: you can use numerical solvers to get an approximate solution)