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(0,1)where1 >1

Denote by p0 = P{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 (x) = 1 {|x| }. Express the FPR and TPR as a function of . Sketch the ROC curve for various values of 1 > 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 k from part 1 that corresponds to this decision rule for p0 = 1/3, 1 = 2. (Hint: you can use numerical solvers to get an approximate solution)