Miss rate when the signal sits closer to the noise

Question

A monitor reads a value X. With no event (H0), X \sim 	ext{Normal}(0, 1). With an event (H1), X \sim 	ext{Normal}(2, 1). The rule declares "event" whenever X 1.5. What is the probability of a type II error (missing a real event)?

Accepted Answer

A type II error occurs when an event is present but X \le 1.5. Compute \beta. Under H1, X \sim 	ext{Normal}(2, 1): \beta = P(X \le 1.5 \mid H1) = \Phi\!\left(\frac{1.5 - 2}{1}ight) = \Phi(-0.5) \approx 0.309. So the rule misses a real event about 31% of the time; power is only about 0.69. Because the signal mean (2) is closer to the noise mean (0) than in the base case, the two distributions overlap more and misses are more common. For context. The false-alarm rate is \alpha = 1 - \Phi(1.5) \approx 0.067. Takeaway. When the signal mean sits closer to the noise, the distributions overlap mo…

Miss rate when the signal sits closer to the noise

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