Th job for i 1 10 then we are trying to find the

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th job, for i ∈ { 1 , . . . , 10 } . Then we are trying to find the probability that X = J 1 + · · · + J 10 is less than 40. We assume that the sum is approximately normal. Note that E ( X ) = 10 E ( J 1 ) = 50 and SD ( X ) = 10 SD ( J 1 ) = 2 10. So P ( X < 40) = P X - 50 2 10 < 40 - 50 2 10 Φ 40 - 50 2 10 = 1 - Φ 10 2 ! 5 . 7% . 8. Assume that in a class of 80 students, each student independently has a 1% chance of forgetting to take the exam. (a) [3 points] What is the probability that two students forget? Solution: Using the binomial (80 , 1 / 100) distribution, 80 2 1 100 2 99 100 78 14 . 43% . (b) [7 points] Use the Poisson approximation to approximate the probability that two students forget. Solution: The expected number of students who forget the exam is μ = np = 0 . 8. So the probability that two students forget is approximately e - 0 . 8 (0 . 8) 2 2! 14 . 38% . Page 5
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Christopher Reinemann
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