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Unformatted text preview: e that the logarithm of the
threshold ratio is given by
ln Λ( (m)) =
y m
Λ(yn); n=1 fY H (yn0) Λ(yn) = ln fY H (yn1) Note that Λ(yn) is a realvalued function of yn, and is the
same function for each n. Thus, since Y1, Y2, . . . , are I ID
rv’s conditional on H = 0 (or H = 1), Λ(Y1), Λ(Y2), are
also I ID conditional on H = 0 (or H = 1).
It follows that ln Λ( (m)), conditional on H = 0 ( or H = 1)
y
is a sum of m I ID rv’s and {ln Λ( (m)); m ≥ 1} is a random
y
walk conditional on H = 0 (or H = 1). The two random
walks contain the same sequence of sample values but
diﬀerent probability measures.
Later we look at sequential detection, where observa
tions are made until a treshold is passed.
7 Threshold tests and the error curve
A general hypothesis testing rule (a test) consists of
mapping each sample sequence into either 0 or 1. Thus
y
a test can b e viewed as the set A of sample sequences
mapped into hypothesis 1. The error probability, given
H = 0 or H = 1, using test A, is given by
q0(A) = Pr {Y ∈ A  H = 0} ; q1(A) = Pr {Y ∈ Ac  H = 1} With a priori probabilities p0, p1 and η = p1/p0,
Pr {e(A)} = p0q0(A) + p1q1(A) = p0[q0(A) + η q1(A)]
For the threshold test based on η ,
Pr {e(η )...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.
 Fall '11
 staff

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