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Unformatted text preview: 1 is true
declare H0 is true. 3.10. BINARY HYPOTHESIS TESTING WITH CONTINUOUS-TYPE OBSERVATIONS
π1 , In particular, the MAP rule is obtained by letting τ =
> γM AP
< γM AP X 115 and it becomes: declare H1 is true
declare H0 is true. 2 π0
where γM AP = m1 −m0 ln π1 + m0 +m1 .
For this example, both the ML and MAP rules have the form X > γ declare H1 is true
< γ declare H0 is true. Therefore, we shall examine the error probabilities for a test of that form. The error probabilities
are given by the areas of the shaded regions shown in Figure 3.25.
" f p
false alarm p
miss 0 m 0 ! f 1 m1 Figure 3.25: Error probabilities for direct threshold detection between two normal pdfs with the
pfalse alarm = P (X > γ |H0 )
γ − m0
X − m0
γ − m0
pmiss = P (X < γ |H1 )
X − m1
γ − m1
m1 − γ
pe = π0 pfalse alarm + π1 pmiss .
Substituting in γ = γM L =
example satisfy: m0 +m1
2 yields that the error probabili...
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