Moss lecture iv fall 2010 d multivariate uniform

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Unformatted text preview: σn2 · · · σnn ￿ (16) C. Univariate uniform distribution f (x|a, b) = ￿∞ ￿1 1 b − a if x ∈ [a, b] 0 otherwise ￿b ￿∞ 1 f (x|a, b) dx = 0dx + dx + 0dx −∞ −∞ a b−a b ￿b ￿ x￿ b a b−a ￿= = − = =1 ￿ b−a a b−a b−a b−1 4 (17) (18) AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture IV Fall 2010 D. Multivariate uniform distribution f ( x 1 , x2 ) = ￿ 1￿ 1 0 0 ￿ 1 if 0 ≤ x1 ≤ 1 and 0 ≤ x2 ≤ 1 0 otherwise f (x1 , x2 ) dx1 dx2 = = ￿1 0 ￿ 1￿ 0 ( x| 1 0 ￿ dx2 = ￿1 0 (1 − 0) dx2 dx2 = (x|1 = 1 − 0 = 1 0 (19) (20) E. To examine the conditional properties of the bivariate uniform distribution, we start by deriving the marginal distribution of x1 . This marginal distribution is derived by integrating out x2 f 1 ( x1 ) = ￿1 0 f (x1 , x2 ) dx2 = ￿1 0 dx2 = (x2 |1 = 1 0 (21) F. The conditional distribution for x2 given the value of x1 is then written as f ( x2 | x1 ) = f ( x 1 , x2 ) 1 = =1 f 1 ( x1 ) 1 (22) Therefo...
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