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
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbffa73e6-1761-4c9e-97cb-0c164f013989%2Fc376e0a1-c606-4d43-b29d-c5e7ac96f723%2Fxsfag99_processed.png&w=3840&q=75)
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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 4 images

Recommended textbooks for you

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON

The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman

Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman