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Unformatted text preview: tations, we express the pdfs without using absolute value signs: f1 (u) = 1 u−1 2e 1 −u+1 2e : u<1 : u≥1 1 u+1 2e 1 −u−1 2e f0 (u) = : u < −1 : u ≥ −1 Therefore, Λ(u) = e−|u−1| = e−|u+1| eu−1 eu+1 = e−2 : eu−1 e−u−1 = e2u e−u+1 e−u−1 = u < −1 : −1 ≤ u < 1 e2 : 1<u The likelihood ratio Λ(u) is nondecreasing and it crosses 1 at u = 0. Thus, the ML decision rule is to decide H1 is true if X > 0 and decide H0 otherwise. The error probabilities for the ML decision rule are: ∞ ∞ −u−1 e 1 pf alse alarm = f0 (u)du = du = ≈ 0.1839. 2 2e 0 0 1 By symmetry, or by a similar computation, we see that pmiss is also given by pmiss = 2e . Of course 1 the average error probability for the ML rule is also 2e for any prior distribution. The MAP decision rule is to choose H1 if Λ(X ) > π0 = 2, and choose H0 otherwise. Note that π1 the solution of e2u = 2 is u = ln 2 , which is between -1 and 1. Thus, the MAP decision rule is 2 √ to choose H1 if X ≥ γM AP and choose H0 otherwise, where γM AP = ln 2 = ln 2. For the MAP 2 decision rule: ∞ pf alse alarm = ln...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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