MIT6_262S11_lec21

# Thus since y1 y2 are i id rvs conditional on h

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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 (yn|0) Λ(yn) = ln fY |H (yn|1) Note that Λ(yn) is a real-valued 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.

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