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3. a) Prove that E[A"(x)l H,] = E[A®™'(x)| H.]. where El·T:] represents a conditional expectation and A"(x) denotes the likelihood ratio raised (LRT) to the nth power . H, and H, represent hypothesis one or null (samples are from class d, or d.) b) The identity you proved in part (a) can be written equivalently as so, choosing n= 0, we know that PATH,) = Ap(ATH,). We also know that FA where T is the threshold in the LRT.

3. a) Prove that E[A"(x)l H,] = E[A®™'(x)| H.]. where El·T:] represents a conditional expectation and A"(x) denotes the likelihood ratio raised (LRT) to the nth power . H, and H, represent hypothesis one or null (samples are from class d, or d.) b) The identity you proved in part (a) can be written equivalently as so, choosing n= 0, we know that PATH,) = Ap(ATH,). We also know that FA where T is the threshold in the LRT.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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3. a) Prove that E[A^(x)I H.] = E[A™*(x)I H.]. n+1 where E[·I-] represents a conditional expectation and A"(x) denotes the likelihood ratio raised (LRT) to the nth power . H, and H, represent hypothesis one or null (samples are from class d, or d.) b) The identity you proved in part (a) can be written equivalently as n+1 so, choosing n= 0, we know that KATH,) = Ap(ATH,). We also know that Po FA where T is the threshold in the LRT. Given these facts and the fact that T20 (it must be since A(x)2 0) show that: i) the slope of the ROC curve at any operating point (P,, PEA)is just the threshold T required to achieve that point, and ii) the ROC curve is nondecreasing.
Transcribed Image Text:3. a) Prove that E[A^(x)I H.] = E[A™*(x)I H.]. n+1 where E[·I-] represents a conditional expectation and A"(x) denotes the likelihood ratio raised (LRT) to the nth power . H, and H, represent hypothesis one or null (samples are from class d, or d.) b) The identity you proved in part (a) can be written equivalently as n+1 so, choosing n= 0, we know that KATH,) = Ap(ATH,). We also know that Po FA where T is the threshold in the LRT. Given these facts and the fact that T20 (it must be since A(x)2 0) show that: i) the slope of the ROC curve at any operating point (P,, PEA)is just the threshold T required to achieve that point, and ii) the ROC curve is nondecreasing.
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