5 aeb 6182 agricultural risk analysis and decision

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Unformatted text preview: re we conclude that x1 and x2 are independent since f 2 ( x2 ) = f ( x2 | x1 ) = 1 (23) G. Univariate Gamma distribution ￿ ￿ 1 x xα−1 exp − β for 0 < x < ∞ and β > 0 Γ (α ) β α f ( x| α , β ) = 0 otherwise (24) III. Transformation of Random Variables A. Another alternative is to create a new distribution by transforming one random variable into another with a known distribution. 5 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture IV Fall 2010 Figure 1: Triangular Distribution Function 1. Nested probability functions F ( x∗ ) = ￿ x∗ −∞ f (x) dx ⇔ P [x ≤ x∗ ] z = φ (x) ⇒ x = ￿φ−1 (z ) ￿ ￿ ￿ ￿ d φ −1 ( z ) ￿ ￿ ￿ g ( z ) = f φ −1 ( z ) ￿ ￿ ￿ dz ￿ (25) (26) 2. Inverse Hyperbolic Sine Distribution ￿ ￿ 2 ln θ￿t + (θ￿t ) + 1 e t = e t ( ￿t , θ ) = ￿1 ￿ 2 θ ￿ ￿ 1 1 ( e t ( ￿ t , θ ) − µ) 2 2 g ￿t |µ, σ , θ = √ exp − 2￿ σ2 σ 2π ￿ ×...
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