06 1 x 2 probability by bayes formula p h1 x 2 p

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Unformatted text preview: pe of error, which is called a “type II error” in the statistics literature. 56 CHAPTER 2. DISCRETE-TYPE 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.

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