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 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 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) 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)

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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

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.

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.
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)
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)