Given that a1 a2 an are exhaustive or a1 a2 an

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Unformatted text preview: hus, the denominator of Equation 7 becomes P [A1 ∩ E ]+P [A2 ∩ E ]+· · · P [An ∩ E ] = P [(A1 ∪ A2 ∪ · · · An ) ∩ E ] (11) 3. Given that A1 , A2 , · · · An are exhaustive, or A1 ∪ A2 ∪ · · · An = C the entire sample space since P [A1 ∪ A2 ∪ · · · An ] = 1 P [(A1 ∪ A2 ∪ · · · An ) ∩ E ] = P [C ∪ E ] = P [E ] with the last equality since E ⊂ C . 3 (12) AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture IV Fall 2010 4. Returning to the numerator in Equation 7 P [E ] = n ￿ i=1 P [E |Ai ] P [Ai ] (13) 5. Definition 2.11 p-20 Two events A and B are independent if P [A] = P [A|B ]. a. In die example P [x1 = 4] = P [x1 = 4|x2 = 6] = 1 6 (14) II. Some Useful Distributrion Functions A. Univariate normal distribution ￿ f x|µ, σ 2 ￿ ￿ 1 ( x − µ) 2 = √ exp − 2σ 2 σ 2π ￿ (15) B. Multivariate normal distribution ￿ 1 1 f (x|µ, Σ) = √ |Σ|−1/2 exp − (x − µ)￿ Σ−1 (x − µ) 2 2π σ11 σ12 · · · σ1n σ21 σ22 · · · σ2n σij Σ= . . .. . ρij √ . . σii σjj . . . . . σn1...
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This note was uploaded on 02/01/2012 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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