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Unformatted text preview: tations, we express the pdfs without using absolute value signs:
f1 (u) = 1 u−1
2e
1 −u+1
2e : u<1
: u≥1 1 u+1
2e
1 −u−1
2e f0 (u) = : u < −1
: u ≥ −1 Therefore, Λ(u) = e−u−1
=
e−u+1 eu−1
eu+1 = e−2 : eu−1
e−u−1 = e2u e−u+1
e−u−1 = u < −1 : −1 ≤ u < 1 e2 : 1<u The likelihood ratio Λ(u) is nondecreasing and it crosses 1 at u = 0. Thus, the ML decision rule is
to decide H1 is true if X > 0 and decide H0 otherwise. The error probabilities for the ML decision
rule are:
∞
∞ −u−1
e
1
pf alse alarm =
f0 (u)du =
du =
≈ 0.1839.
2
2e
0
0
1
By symmetry, or by a similar computation, we see that pmiss is also given by pmiss = 2e . Of course
1
the average error probability for the ML rule is also 2e for any prior distribution.
The MAP decision rule is to choose H1 if Λ(X ) > π0 = 2, and choose H0 otherwise. Note that
π1
the solution of e2u = 2 is u = ln 2 , which is between 1 and 1. Thus, the MAP decision rule is
2
√
to choose H1 if X ≥ γM AP and choose H0 otherwise, where γM AP = ln 2 = ln 2. For the MAP
2
decision rule:
∞ pf alse alarm =
ln...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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