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Optional_HW2sol

# Optional_HW2sol - 3 By central limit theorem S n-nμ √...

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ISyE 2027B Probability with Applications Fall 2010 Optional Homework 2 Solutions 1. We want to find n such that P ( | ¯ U n | < 0 . 5) 0 . 9, by Chebyshev’s inequal- ity, P ( | ¯ U n | ≥ 0 . 5) V ar ( U i /n ) 0 . 5 2 = 12 n 1 - 0 . 9 = 0 . 1 Thus n 120. 2. By central limit theorem, S n - nμ/ N (0 , 1), n = 144, μ = 2, σ = 2, P ( S n > 264) = P ( S n - > 264 - 144 × 2 144 × 2 ) = P ( Z n > - 1) = 0
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Unformatted text preview: 3. By central limit theorem, S n-nμ/ √ nσ ⇒ N (0 , 1), n = 10 , 000, μ = 240, σ = 800, P ( S n > 2 . 7 × 10 6 ) = P ( S n-nμ √ nσ > 2 . 7 × 10 6-10 , 000 × 240 √ 10 , 000 × 800 ) = P ( Z n > 3 . 75) = 0 . 0001 1...
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