Optimize thresholds under fraud costs

Cost-sensitive thresholding: You have a binary ATO classifier that scores transfers. The initial prevalence (fraud rate) is 0.2%. The costs are: a false positive (blocking a legitimate transfer) costs $2; a false negative (letting a fraudulent transfer through) costs $120. Consider the following operating points measured on a large validation set:

Assume 1,000,000 transfers are evaluated.

Tasks: A) For each threshold, compute the expected total cost as $\text{FP} \times \$2 + \text{FN} \times \$120$ given a prevalence of 0.2%. Choose the threshold that minimizes cost, and report PPV and NPV at that operating point. B) If prevalence falls to 0.1% because of seasonality, recompute the costs and discuss whether the optimal threshold changes. C) Using ROC theory, derive the slope of the cost-optimal decision rule, $\lambda = \frac{C_{fp}}{C_{fn}} \cdot \frac{1-\pi}{\pi}$, and explain how it maps to selecting a point on the ROC curve; interpret how prevalence shifts move the optimal operating point without retraining.

Overview: This question evaluates a candidate's competency in cost-sensitive binary classification, threshold selection, and ROC-based operating-point analysis, including computing expected costs and predictive values under varying prevalence and asymmetric error costs.