# B 8 points find an upper bound for p x y 25 solution

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(b) [8 points] Find an upper bound for P ( X + Y 25). Solution: Let μ = E ( X + Y ) and σ = SD ( X + Y ). Using Chebychev’s in- equality, P ( X + Y 25) = P ( X + Y - μ 15) P ( | X + Y - μ | ≥ 15) = P ( | X + Y - μ | ≥ 3 σ ) 1 3 2 = 1 9 . (c) [2 points] Assuming that X and Y are symmetric (hence also X + Y ), improve the answer to part (b). Solution: By symmetry (meaning symmetry around μ ), P ( X + Y - μ 3 σ ) = P ( X + Y - μ ≤ - 3 σ ). Since P ( X + Y - μ 3 σ ) + P ( X + Y - μ ≤ - 3 σ ) = P ( | X + Y - μ | ≥ 3 σ ) 1 9 , each of these probabilities is at most 1 / 18. 7. [10 points] Assume that each job you have lasts an average of 5 years with a standard deviation of 2. Approximate the probability that you will have more than 10 jobs in 40 years. ( Hint : ﬁnd the probability that your ﬁrst 10 jobs last less than 40 years.) Page 4

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Math 431: Exam 1 Solution: Let J i be the length of the your i 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 ² Φ ± 40 - 50 2 10 ² = 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 ²± 1 100 ² 2 ± 99 100 ² 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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b 8 points Find an upper bound for P X Y 25 Solution Let μ...

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