Cost-sensitive thresholding: You have a binary ATO classifier scoring transfers. Prevalence (fraud rate) is 0.2% initially. Costs: false positive (blocking a legit transfer) costs $2; false negative (letting a fraud through) costs $120. Consider the following operating points measured on a large validation set: Threshold | TPR | FPR 0.90 | 0.50| 0.0010 0.80 | 0.65| 0.0030 0.70 | 0.75| 0.0060 0.60 | 0.82| 0.0100 0.50 | 0.88| 0.0180 Assume 1,000,000 evaluated transfers. Tasks: A) For each threshold, compute expected total cost = (FP * $2) + (FN * $120) given prevalence 0.2%. Choose the cost-minimizing threshold and report PPV and NPV at that point. B) If prevalence drops to 0.1% due to seasonality, recompute and discuss whether the optimal threshold changes. C) Using ROC theory, derive the cost-optimal decision rule slope λ = (C_fp/C_fn) * ((1−π)/π) and explain how it maps to choosing a point on the ROC; interpret how prevalence shifts move the optimal operating point without retraining.