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# The following two properties are necessary and

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Unformatted text preview: = KM AP = ab 2 ln(bπ0a2 1 ) . b2 − Finally, we ﬁnd the error probabilities for the magnitude threshold test (3.12) with an arbitrary threshold K &gt; 0. Substituting in KM L or KM AP for K gives the error probabilities for the ML and 3.10. BINARY HYPOTHESIS TESTING WITH CONTINUOUS-TYPE OBSERVATIONS 117 MAP tests: ∞ −K pf alse alarm = P {|X | &gt; K | H0 } = f0 (u)du = Φ(−K/a) + 1 − Φ(K/a) = 2Q(K/a). f0 (u)du + −∞ K pmiss = P {|X | &lt; K | H1 } = K f1 (u)du = Φ(K/b) − Φ(−K/b) = 1 − 2Q(K/b). −K Example 3.10.3 Based on a sensor measurement X , it has to be decided which hypothesis about a remotely monitored room is true: H0 : the room is empty vs. H1 : a person is present in the 1 room. Suppose if H0 is true then X has pdf f0 (x) = 2 e−|x+1| and if H1 is true then X has pdf f1 (x) = 1 e−|x−1| . Both densities are deﬁned on the whole real line. Find the ML decision rule, the 2 MAP decision rule for prior probability distribution (π0 , π1 ) = (2/3, 1/3), and the associated error probabilities, including the average error probability based on the given prior. Solution: To help with the compu...
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