42 joint probability mass functions if x and y are

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Unformatted text preview: 1 is true declare H0 is true. 3.10. BINARY HYPOTHESIS TESTING WITH CONTINUOUS-TYPE OBSERVATIONS π0 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 2σ where γM AP = m1 −m0 ln π1 + m0 +m1 . 2 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 same variance. pfalse alarm = P (X > γ |H0 ) γ − m0 X − m0 > =P H0 σ σ γ − m0 . =Q σ pmiss = P (X < γ |H1 ) X − m1 γ − m1 =P < H1 σ σ m1 − γ =Q . σ 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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