False-alarm rate of a fraud flag threshold

Question

A fraud model outputs a score X. For legitimate transactions (H0), X \sim 	ext{Normal}(50, 10). For fraudulent ones (H1), X \sim 	ext{Normal}(80, 10). The rule flags "fraud" whenever X 70. What is the probability of a type I error (flagging a legitimate transaction)?

Accepted Answer

A type I error means the transaction is legitimate but the score exceeds the cutoff of 70. Compute \alpha. Under H0, X \sim 	ext{Normal}(50, 10): \alpha = P(X 70 \mid H0) = 1 - \Phi\!\left(\frac{70 - 50}{10}ight) = 1 - \Phi(2) \approx 0.023. So about 2.3% of legitimate transactions get flagged. For context, the miss rate. Under H1, X \sim 	ext{Normal}(80, 10): \beta = P(X \le 70 \mid H1) = \Phi\!\left(\frac{70 - 80}{10}ight) = \Phi(-1) \approx 0.159. So about 16% of true fraud slips through. If missed fraud is costlier than the annoyance of a false flag, you would lower the cutoff below…

False-alarm rate of a fraud flag threshold

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