Unformatted text preview: ties for the ML rule for this pfalse alarm = pmiss = pe = Q m1 − m0
2σ . 116 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES Note that m1 −m0 can be interpreted as a signal-to-noise ratio. The diﬀerence in the means, m1 − m0 ,
can be thought of as the diﬀerence between the hypotheses due to the signal, and σ is the standard
deviation of the noise. The error probabilities for the MAP rule can be obtained by substituting
in γ = γM AP in the above expressions. Example 3.10.2 Based on a sensor measurement X , it has to be decided which hypothesis about
a remotely monitored machine is true: H0 : the machine is working vs. H1 : the machine is broken.
Suppose if H0 is true X is normal with mean 0 and variance a2 , and if H1 is true X is normal with
mean 0 and variance b2 . Suppose a and b are known and that 0 < a < b. Find the ML decision rule,
the MAP decision rule (for a given choice of π0 and π1 ) and the error probabilities, pf alse alarm and
pmiss , for both rules.
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