Unformatted text preview: pe of error, which is called a
“type II error” in the statistics literature. 56 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES entry in each column of the likelihood matrix. If the entries in a column of the likelihood matrix
are identical, then either can be underlined. The choice may depend on other considerations such
as whether we wish to minimize pfalse alarm or pmiss . The ML rule for the example likelihood matrix
above is the following: H1
H0 X=0 X=1 X=2 X=3
0.0
0.1
0.3
0.6
0.4
0.3
0.2
0.1 ← underlines indicate
the ML decision rule . It is easy to check that for the ML decision rule, pfalse alarm = 0.2+0.1 = 0.3 and pmiss = 0.0+0.1 = 0.1.
There is another way to express the ML decision rule. Note that for two positive numbers a
and b, the statement a > b is equivalent to the statement that a > 1. Thus, the ML rule can be
b
rewritten in a form called a likelihood ratio test (LRT) as follows. Deﬁne the likelihood ratio Λ(k )
for each possible observation k as the ratio of the two conditional probabilities:
Λ(k ) = p1 (k )
.
p0 (k ) The ML rule is thus equivalent to deciding that H1 is true if Λ(X ) > 1 and dec...
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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 Tech.
 Spring '08
 Zahrn
 The Land

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