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Lecture12

# Lecture12 - Central Limit Theorem A More General Central...

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Central Limit Theorem 11/08/2005

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A More General Central Limit Theorem Theorem. Let X 1 , X 2 , . . . , X n , . . . be a sequence of indepen- dent discrete random variables, and let S n = X 1 + X 2 + · · · + X n . For each n , denote the mean and variance of X n by μ n and σ 2 n , respectively. Define the mean and variance of S n to be m n and s 2 n , respectively, and assume that s n → ∞ . If there exists a constant A , such that | X n | ≤ A for all n , then for a < b , lim n →∞ P a < S n - m n s n < b = 1 2 π Z b a e - x 2 / 2 dx . 1
Midterm Review Let S n be the number of successes in n Bernoulli trials with probability .8 for success on each trial. Let A n = S n /n be the average number of successes. In each case give the value for the limit, and give a reason for your answer. 1. lim n →∞ P ( A n = . 8) . 2. lim n →∞ P ( . 7 n < S n < . 9 n ) . 3. lim n →∞ P ( S n < . 8 n + . 8 n ) . 4. lim n →∞ P ( . 79 < A n < . 81) . 2

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In the middle of the night cars arrive at the I-91 security blockade, just south of White River Junction, at an average rate of 18 cars per hour.
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