231hw2solutionsF11

231hw2solutionsF11 - Will Landau August 21, 2011 STAT 231...

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Unformatted text preview: Will Landau August 21, 2011 STAT 231 Problem Set 2 Solutions Exercise 2.1 (Devore 2.1) . a. If team 1 places first, team 3 places second, team 2 places third, and team 4 places last, then we denote the outcome by 1324. Since the competition is a tournament and teams 1 and 2 play first, either: i. Team 1 beats team 2, so team 1 goes to the championship game and team 2 does not. 1 is among the left two digits and 2 is among the right two digits of the outcome, or: ii. Team 2 beats team 1, so team 2 goes to the championship game and team 1 does not. 2 is among the left two digits and 1 is among the right two digits of the outcome We place a similar constraint on numbers 3 and 4 in the outcome. These constraints leave us with the following outcomes in S : { 1324 , 1342 , 1423 , 1432 , 2314 , 2341 , 2413 , 2431 , 3124 , 3142 , 4123 , 4132 , 3214 , 3241 , 4213 , 4231 } b. A is the subset of S with outcomes having a left-most digit of 1. Hence, A = { 1324 , 1342 , 1423 , 1432 } . c. B is the subset of S with a 2 as one of the left-most two digits. Hence, B = { 2314 , 2341 , 2413 , 2431 , 3214 , 3241 , 4213 , 4231 } . d. i. A ∪ B = outcomes in either A or B or both = { 1324 , 1342 , 1423 , 1432 , 2314 , 2341 , 2413 , 2431 , 3214 , 3241 , 4213 , 4231 } . ii. A ∩ B = ∅ . This makes sense because if team 1 won the tournament, then team 2 wouldn’t have made it to the championship game. iii. A = S - A = { 2314 , 2341 , 2413 , 2431 , 3124 , 3142 , 4123 , 4132 , 3214 , 3241 , 4213 , 4231 } Exercise 2.2 (Devore 2.3) . a. Be careful about the notation you pick for your outcomes. It’s important to translate the problem into symbols in a way that works. I will denote a success in component 1, a failure in component 2, and a failure in component 3 by the symbol, SFF. Then, A = { SSF,SFS,FSS } b. B = { SSS,SSF,SFS,FSS } c. C = { SSS,SSF,SFS } , since component 1 must function and at least one of components 2 and 3 must function. 1 d. C = { SFF,FSS,FSF,FFS,FFF } A ∪ C = { SSS,SSF,SFS,FSS } A ∩ C = { SSF,SFS } B ∪ C = { SSS,SSF,SFS,FSS } B ∩ C = { SSSSSF,SFS } Exercise 2.3 (Devore 2.4) . a. Pay attention to the pattern I used to construct this table. Outcome Mort. 1 Mort. 2 Mort 3. Mort. 4 1 F F F F 2 F F F V 3 F F V F 4 F F V V 5 F V F F 6 F V F V 7 F V V F 8 F V V V 9 V F F F 10 V F F V 11 V F V F 12 V F V V 13 V V F F 14 V V F V 15 V V V F 16 V V V V b. Outcome numbers 2, 3, 5, 9 c. Outcome numbers 1, 16 d. Outcome numbers 1, 2, 3, 5, 9 e. The union is the event that either: i. all of the mortgages are variable, or: ii. at most at most one of them is variableii....
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This note was uploaded on 02/11/2012 for the course STAT 231 taught by Professor Staff during the Fall '08 term at Iowa State.

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231hw2solutionsF11 - Will Landau August 21, 2011 STAT 231...

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