# Joint probability density functions the random

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Unformatted text preview: olution: To begin, we get as simple expression for the likelihood ratio as we can: 2 Λ(u) = u 1 √ e− 2b2 b 2π 1 √e a 2π 2 − u2 2a = a − u2 + u22 a u2 1 1 e 2b2 2a = e 2 ( a2 − b2 ) b b The ML rule is to choose H1 when Λ(X ) > 1. Thus, by taking the natural logarithm of both sides 2 1 of this inequality we obtain the rule: If (ln a ) + X ( a2 − b1 ) > 0 choose H1 . Equivalently, after a 2 b 2 bit of algebra, we ﬁnd that the ML rule selects H1 when ln b X 2 b2 − a2 < a 2 a2 b2 or 2a2 b2 ln(b/a) b2 − a2 < X 2. Thus, the ML rule can be expressed as a threshold test on the magnitude |X | of X : |X | > K declare H1 is true < K declare H0 is true. (3.12) ln(b/a where K = KM L = ab 2b2 −a2 ) . π0 The MAP rule is to choose H1 when Λ(X ) > π1 . After a bit of algebra, we derive the rule that H1 should be chosen when bπ0 X 2 b2 − a2 ln < aπ1 2 a2 b2 Or, equivalently, the MAP rule is given by the magnitude threshold test (3.12) with the threshold /aπ K...
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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.

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