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elemprob-page27

# elemprob-page27 - Answer We want P Sn 3.5 >.05 n We rewrite...

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Answer. We want P S n n - 3 . 5 > . 05 . We rewrite this as P ( | S n - n E X 1 | > ( . 05)(3600)) = P S n - n E X 1 n Var X 1 > 180 (60) 35 12 P ( | Z | > 1 . 756) . 08 . Example. Suppose the lifetime of a human has expectation 72 and variance 36. What is the probability that the average of the lifetimes of 100 people exceeds 73? Answer. We want P S n n > 73 = P ( S n > 7300) = P S n - n E X 1 n Var X 1 > 7300 - (100)(72) 100 36 P ( Z > 1 . 667) . 047 . The idea behind proving the central limit theorem is the following. It turns out that if m Y n ( t ) m Z ( t ) for every t , then P ( a Y n b ) P ( a Z b ). (We won’t prove this.) We are going to let Y n = ( S n - ) n . Let W i = ( X i - μ ) . Then E W i = 0, Var W i = Var X i σ 2 = 1, the W i are independent, and S n - σ n = n i =1 W i n . So there is no loss of generality in assuming that μ = 0 and σ = 1. Then m Y n ( t ) = E e tY n = E e ( t/ n )( S n ) = m S n ( t/
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