131A_1_ds7_2010fall_Sol

131A_1_ds7_2010fall_Sol - EE 131A Probability Instructor:...

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EE 131A Discussion Set 7 Probability Wednesday, November 10, 2010 Instructor: Lara Dolecek and Friday, November 12, 2010 Reading: Chapters 5 of Probability, Statistics, and Random Processes by A. Leon-Garcia 1. Coin tossing and two RVs. Problem 5.1, page 288 of ALG (a) In this problem, let X be the maximum number and Y be the minimum number of heads. S ( Carlos,Michael ) = { (0 , 0) , (0 , 1) , (0 , 2) , (1 , 0) , (1 , 1) , (1 , 2) , (2 , 0) , (2 , 1) , (2 , 2) } S XY ( Y,X ) = { (0 , 0) , (0 , 1) , (0 , 2) , (1 , 1) , (1 , 2) , (2 , 2) } The mapping is shown as the following: S (0 , 0) S XY (0 , 0) S (1 , 1) S XY (1 , 1) S (2 , 2) S XY (2 , 2) S (0 , 1) ,S (1 , 0) S XY (0 , 1) S (0 , 2) ,S (2 , 0) S XY (0 , 2) S (1 , 2) ,S (2 , 1) S XY (1 , 2) (b) P (0 , 0) = P (2 , 2) = 1 4 × 1 4 = 1 16 P (1 , 1) = ( 2 1 ) × 1 4 × ( 2 1 ) × 1 4 = 1 4 P (0 , 1) = P (1 , 2) = 2 × 1 4 × ( 2 1 ) × 1 4 = 1 4 P (0 , 2) = 2 × 1 4 × 1 4 = 1 8 (c) P ( X = Y ) = P (0 , 0) + P (1
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131A_1_ds7_2010fall_Sol - EE 131A Probability Instructor:...

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