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Unformatted text preview: W and Z are uncorrelated) which, since they are jointly Gaussian, makes them independent. Problem 5. (a) Assuming hypothesis H 1 is twice more likely than hypothesis H , then P ( H ) = 1 3 , P ( H 1 ) = 2 3 . τ = P ( H ) P ( H 1 ) = 1 2 . Therefore, L ( x ) = P ( x  H 1 ) P ( x  H ) H 1 > < H 1 2 . Plugging in the value, we have L ( x ) =  x  4 1 4 H 1 > < H 1 2 = ⇒  X  H 1 > < H 1 2 . (b) P (error) = P (error  H was true) P ( H ) + P (error  H 1 was true) P ( H 1 ) IE 300 / GE 331: Midterm Exam Solution 3 P (error  H was true) = Z  x  > 1 2 1 4 d x = 2 Z 2 1 2 1 4 d x = 2 · 1 4 · (21 2 ) = 3 4 . P (error  H 1 was true) = Z  x  < 1 2  x  4 d x = 2 Z 1 2 x 4 d x = 2 · x 2 8 ± ± ± ± ± 1 2 = 1 4 · ( 1 40) = 1 16 . = ⇒ P (error) = 3 4 · 1 3 + 1 16 · 2 3 = 1 4 + 1 24 = 7 24 ....
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 Spring '09
 NegarKayavash
 Normal Distribution, Variance, 2W, 100 days

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