homework5sol

homework5sol - ⇒ X I ∼ Bin(9 1 and X G ∼ Bin(9 9 1(a...

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Homework 5 Solution Note: Normally, I will assign 10 points to each question. There are 9 questions plus one bonus question in homework 4. So, the total points are 90+10. If you have any questions related to the homework points or solutions, I am available through email [email protected] and ANGEL. However, DO NOT turn in your homework to me directly unless you have the instructor’s permisson. Exercises 3.4, Question 2 X Bin (15 , 0 . 3) (a) (i) binomial (b) EX = (0 . 3) * 15 = 4 . 5, V arX = (0 . 3) * (1 - 0 . 3) * 15 = 3 . 15 (c) > dbinom(6,15,0.3) [1] 0.147236 > 1-pbinom(5,15,0.3) [1] 0.2783786 P ( X = 6) = 0 . 1472 , P ( X 6) = 1 - P ( X < 6) = 1 - P ( X 5) = 0 . 2784 Exercises 3.4, Question 4 X Bin (0 . 5 , 10) (a) EX = (0 . 5) * 10 = 5, V arX = (0 . 5) * (0 . 5) * 10 = 2 . 5 (b) P ( X = 5) = 10 5 (0 . 5) 5 (0 . 5) 5 0 . 246 (c) P ( X = 0) + P ( X = 1) + P ( X = 2) + P ( X = 3) + P ( X = 4) + P ( X = 5) = 0 . 623 (d) Y is the number of incorrectly answered questions. (e) P (2 Y 5) = P (2 10 - X 5) = P (5 X 8) = 0 . 6123 . Exercises 3.4, Question 6 Let X I = the number of judges that vote guilty conditioning on the defendant is inno- cent Let X G = the number of judges that vote guilty conditioning on the defendant is guilty

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Unformatted text preview: ⇒ X I ∼ Bin (9 , . 1) and X G ∼ Bin (9 , . 9) 1 (a) P (defendant convicted) = P (convicted | guilty) P (guilty) + P ( convicted | innocent ) P (innocent) = P ( X G ≥ 5) P ( guilty ) + P ( X I ≥ 5) P ( innocent ) = [1-P ( X G ≤ 4)](0 . 6) + [1-P ( X I ≤ 4)](0 . 4) = 0 . 5998218 (b) P (correct verdict) = P (correct | guilty) P (guilty) + P (correct | innocent) P (innocent) = P ( X G ≥ 5) P ( guilty ) + P ( X I ≤ 4) P ( innocent ) = [1-P ( X G ≤ 4)](0 . 6) + P ( X I ≤ 4)(0 . 4) = 0 . 999109 Exercises 3.4, Question 8 (a) (iii) negative binomial distribution with r = 5 and p = 0 . 05. (b) S x = { 5 , 6 ,... } . The pmf of X is given by : P ( X = x ) = ± x-1 5-1 ² (0 . 05) 5 (0 . 95) x-5 , x = 5 , 6 ,.... (c) P ( X > 35) = 1-P ( X ≤ 35) = 0 . 9709737 by R command below. > 1-pnbinom(30,5,0.05) [1] 0.9709737 2 3 4...
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