The following two properties are necessary and

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 > 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 | > K | H0 } = f0 (u)du = Φ(−K/a) + 1 − Φ(K/a) = 2Q(K/a). f0 (u)du + −∞ K pmiss = P {|X | < 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...
View Full Document

Ask a homework question - tutors are online