20061ee131A_1_131a2sol

20061ee131A_1_131a2sol - EE 131A Homework#2 Solution Winter...

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EE 131A Homework #2 Winter 2006 Solution K. Yao 1. a. S = set of ordered pairs ( x, y ) where x, y ∈ { 1 , 2 , 3 , 4 , 5 , 6 } . 1 2 3 4 5 6 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) b. { (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) , (6 , 6) } . c. { (1 , 1) , (1 , 2) , (2 , 1) , (2 , 3) , (3 , 2) , (1 , 4) , (4 , 1) , (6 , 1) , (1 , 6) , (5 , 2) , (2 , 5) , (3 , 4) , (4 , 3) , (5 , 6) , (6 , 5) } . 2. a. S = { (1 , 2 , 3) , (2 , 1 , 3) , (3 , 1 , 2) , (1 , 3 , 2) , (2 , 3 , 1) , (3 , 2 , 1) } b. A 1 = { (1 , 2 , 3) , (1 , 3 , 2) } , A 2 = { (1 , 2 , 3) , (3 , 2 , 1) } , A 3 = { (1 , 2 , 3) , (2 , 1 , 3) } . c. A 1 A 2 A 3 = { (1 , 2 , 3) } ≈ “the number of each ball corresponds to the number of the draw.” d. A 1 A 2 A 3 = { (1 , 2 , 3) , (1 , 3 , 2) , (3 , 2 , 1) , (2 , 1 , 3) } ≈ “at least one number of the ball corresponds to the number of the draw.” e. ( A 1 A 2 A 3 ) c = { (3 , 1 , 2) , (2 , 3 , 1) } ≈ “none of the balls has a number corresponding to the number of the draw.” 3. I [ - 1 , 1]. P [ I ] = k × length( I ). 1 = P [[ - 1 , 1]] = k × length([ - 1 , 1]) = k × 2. k = 1 2 . P [ I ] = 1 2 × length( I ). a. A = {- 1 x < 0 } , P [ A ] = 1 2 × 1 = 1 2 ; B = {| x - 0 . 5 | < 1 } = {- 0 . 5 < x 1 } , P [ B ] = 1 2 × 3 2 = 3 4 . C = { x > 0 . 75 } = { 0 . 75 < x 1 } , P [ C ] = 1 2 × 1 4 = 1 8 .
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