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 \lambda_0=1 (hypothesis H_0) or with the average rate \lambda_1=\lambda_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_{md}=C_{01} is ten times the cost of false alarm C_{fa}=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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思路分析
We start by restating the problem in our own words to ground the analysis. We have Y ~ Exponential with rate lambda0 = 1 under H0 or with rate lambda1 = 0.1 under H1. Priors: P(H0) = 0.9, P(H1) = 0.1. Costs: C_fa = 10 thousand dollars for a false alarm (decide H1 when H0 is true) and C_md = 100 thousand dollars for a missed detection (decide H0 when H1 is true). Correct decisions cost 0.
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类似问题
The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times Ym between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate λ. A healthy patient is characterized by the rate λ0, while an unhealthy one is characterized by λ1<λ0. M measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of λ0 and λ1 are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed, λ0 with mean value ˉ λ 0, and λ1 with mean value ˉ λ 1. With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to η= P0 P1 Cfa Cmd =1. M=10 observations are made on a patient, yielding the measured values y1,…yM. If ˉ λ 0=1 and ˉ λ 1= ˉ λ 0/4, which of the following represents the Bayesian decision rule (numbers are rounded):
The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate . A healthy patient is characterized by the rate , while an unhealthy one is characterized by . measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of and are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed, with mean value , and with mean value . With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to . observations are made on a patient, yielding the measured values . If and , which of the following represents the Bayesian decision rule (numbers are rounded):
The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times 𝑌 𝑚 between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate 𝜆 . A healthy patient is characterized by the rate 𝜆 0 , while an unhealthy one is characterized by 𝜆 1 < 𝜆 0 . 𝑀 measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of 𝜆 0 and 𝜆 1 are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed, 𝜆 0 with mean value 𝜆 ¯ 0 , and 𝜆 1 with mean value 𝜆 ¯ 1 . With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to 𝜂 = 𝑃 0 𝑃 1 𝐶 𝑓 𝑎 𝐶 𝑚 𝑑 = 1 . 𝑀 = 10 observations are made on a patient, yielding the measured values 𝑦 1 , … 𝑦 𝑀 . If 𝜆 ¯ 0 = 1 and 𝜆 ¯ 1 = 𝜆 ¯ 0 / 4 , which of the following represents the Bayesian decision rule (numbers are rounded):
Random variable Y is known to be exponentially distributed, either with the average rate λ0=1 (hypothesis H0) or with the average rate λ1=λ0/10 (hypothesis H1). A decision H0/H1 has to be made based on a single observation y of Y. The cost of missed detection, Cmd=C01 is ten times the cost of false alarm Cfa=C10=10 thousand dollars. Correct decisions cost nothing. If it is known that P1=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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