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# 02 001 the ml decisions are indicated by the

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Unformatted text preview: binomial coeﬃcients cancel, so Λ(k ) = 2k 3 1 5−k 3 15 2 2 3 k =2 5 ≈ 2k . 7.6 Therefore, the ML decision rule is to declare H1 whenever Λ(X ) ≥ 1, or equivalently, X ≥ 3. For the ML rule, 1 2 pfalse alarm = 10 5 +5 5 1 2 + 1 2 5 15 2 14 22 +5 + 10 3 3 3 3 = (0.2)pfalse alarm + (0.8)pmiss ≈ 0.26. pmiss = pe = 0.5 1 3 3 = 51 ≈ 0.201 243 The MAP decision rule is to declare H1 whenever Λ(X ) ≥ 0.25, or equivalently, X ≥ 1. That is, the MAP rule declares H0 only if X = 0. For the MAP rule, pfalse alarm = 1 − 1 2 5 ≈ 0.97 15 1 = ≈ 0.041 3 243 = (0.2)pfalse alarm + (0.8)pmiss ≈ 0.227. pmiss = pe As expected, pe for the MAP rule (designed with the correct prior probabilities in mind) is smaller than pe for the ML rule. 2.11. BINARY HYPOTHESIS TESTING WITH DISCRETE-TYPE OBSERVATIONS 59 Example 2.11.2 (Detection problem with Poisson distributed observations) A certain deep space transmitter uses on-oﬀ modulation of a laser to send a bit, with value either zero or one. If the bit is zero, the number of photons, X, arriving at the re...
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