HW5solution_1 - Econ 3190 Fall 2011 HW#5 Solution 1....

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Econ 3190 Fall 2011 HW#5 Solution 1. Bivariate Discrete R.V. (a) X X X Y f ( x;y ) = f (1 ; 1) + f (1 ; 2) + f (1 ; 3) + f (2 ; 1) + f (2 ; 2) + f (2 ; 3) + f (3 ; 1) + f (3 ; 2) + f (3 ; 3) = c (1 + 2 + 3 + 2 + 4 + 6 + 3 + 6 + 9) = 36 c = 1 ) c = 1 36 Also, f ( x;y ) = xy 36 & 0 ; for x = 1 ; 2 ; 3; y = 1 ; 2 ; 3 : (b) X X X Y f ( x;y ) = c [ 2 + 2 j + 2 ± 3 j + j 0 + 2 j + j 0 ± 3 j + j 2 + 2 j + j 2 + 3 j = 15 c = 1 ) c = 1 15 : Also, f ( x;y ) = 1 15 j x ± y j & 0 ; for x = ± 2 ; 0 ; 2 ; y = ± 2 ; 3 : 2. Using pdf to &nd probability of bivariate DRV; (a) P ( X ² 2 ;Y = 1) = f (0 ; 1) + f (1 ; 1) + f (2 ; 1) = 0 + 1 + 1 + 1 + 2 + 1 30 = 0 : 2 (b) P ( X > 2 ;Y ² 1) = f (3 ; 0) + f (3 ; 1) = 3 + 0 + 3 + 1 30 = 0 : 23 1
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(c) P ( X > Y ) = f (1 ; 0) + f (2 ; 0) + f (2 ; 1) + f (3 ; 0) + f (3 ; 1) + f (3 ; 2) = 1 + 2 + 2 + 1 + 3 + 3 + 1 + 3 + 2 30 = 0 : 6 (d) P ( X + Y = 4) = f (2 ; 2) + f (3 ; 1) = 2 + 2 + 3 + 1 30 = 0 : 27 3. Marginal density for C.R.V. (a) f X ( x ) = Z 1 0 2 3 ( x + 2 y ) dy = 2 3 ( xy + y 2 ) j 1 0 = 2 3 ( x + 1) for 0 x 1 Also f X ( x ) = 0 elsewhere. (b)
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HW5solution_1 - Econ 3190 Fall 2011 HW#5 Solution 1....

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