题目
题目

EECE5612.MERGED.202530 quiz 2

单项选择题

Random variable 𝑌  is known to be exponentially distributed, either with the average rate 𝜆 0 = 1  (hypothesis 𝐻 0 ) or with the average rate 𝜆 1 = 𝜆 0 / 10  (hypothesis 𝐻 1 ). A decision 𝐻 0 / 𝐻 1  has to be made based on a single observation 𝑦  of 𝑌 . The cost of missed detection, 𝐶 𝑚 𝑑 = 𝐶 01  is ten times the cost of false alarm 𝐶 𝑓 𝑎 = 𝐶 10 = 10 thousand dollars. Correct decisions cost nothing. If it is known that 𝑃 1 = 0.1 , specify the Bayesian decision rule that will minimize the average cost. What is the resulting average cost? Mark the correct answer (numbers are rounded):

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We start by restating the setup and what we are minimizing. - Y is exponential under H0 with rate λ0 = 1, so f0(y) = e^{-y}, y ≥ 0. - Y is exponential under H1 with rate λ1 = λ0/10 = 0.1, so f1(y) = 0.1 e^{-0.1 y}, y ≥ 0. - Prior: P(H0) = P0 = 0.9, P(H1) = P1 = 0.1. - Costs: false alarm (decide H1, but H0 true) costs C01 = 100,000; miss (decide H0, but H1 true) costs C10 = 10,000. The problem states CMD = C01 is ten times the cost of a false alarm CAD? incorrectly phrased in my notes, but the intended setup is: C01 = 10 × C10, so C10 = 10,000 and C01 = 100,000. - We seek the Bayesian decision rule that minimizes Bayes risk, and then the resulting average cost under that rule. Step-by-s......Login to view full explanation

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