hw9 - Math 3C Homework 9 Solutions Ilhwan Jo and Akemi...

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Unformatted text preview: Math 3C Homework 9 Solutions Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 12.6 Problems 15, 16, 17(a), 18(a), 19(a), 20(a), 23-26 15. Toss a fair coin 400 times. Use the central limit theorem to find an approximation for the probability of at most 190 heads. Solution Let X i = ( 1 , if i th toss is heads , otherwise . Since the X i are i.i.d. and binomially distributed we know = EX i = np = (1) 1 2 = 1 2 , 2 = var( X i ) = np (1- p ) = (1) 1 2 1- 1 2 = 1 4 . Now let S 400 = 400 i =1 X i count the number of heads we get out of 400 coin tosses. By the central limit theorem we get P ( S 400 190) = P S 400- 400 400 2 190- 200 10 = P S 400- 400 400 2 - 1 1- (1) = 1- . 8413 = 0 . 1587 16. Toss a fair coin 150 times. Use the central limit theorem to find an approximation for the probability that the number of heads is at least 70. Solution Let X i be defined as in the previous problem and let S 150 = 150 i =1 X i . By the central limit theorem P ( S 150 70) = P S 150- 150 150 2 70- 75 37 . 5 = 1- P S 150- 150 150 2 - . 82 1- (- . 82) = (0 . 82) = 0 . 7939 1 17a. Toss a fair coin 200 times. Use the central limit theorem to find an approximation for the probability that the number of heads is at least 120. Solution Let X i be defined above and let S 200 = 200 i =1 X i . By the central limit theorem P ( S 200 120) = P S 200- 200 200 2 120- 100 50 = 1- P S 200- 200 200 2...
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hw9 - Math 3C Homework 9 Solutions Ilhwan Jo and Akemi...

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