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Section9Notes - SECTION NOTE 9 ORIE360/560 FALL 2004 CHONG...

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SECTION NOTE 9 ORIE360/560, FALL 2004 CHONG WANG 1. Central Limit Theorem Let X 1 ,X 2 ,.... ,X n be i.i.d. with EX i = μ , V ar ( X i ) = σ 2 . If S n = X 1 + ... + X n , then S n - σn 1 / 2 χ where χ has standard normal distribution. Here ” ” means ”converges in distribu- tion”. In other words, if let G n ( x ) denote the cdf of ( S n - ) / ( σn 1 / 2 ). Then for any x , -∞ < x < , lim n →∞ G n ( x ) = Z x -∞ 1 2 π e - y 2 / 2 dy that is, ( S n - ) / ( σn 1 / 2 ) has a limiting standard normal distribution. Note: If we define ¯ X n = (1 /n ) n i =1 X i = (1 /n ) S n , then S n - σn 1 / 2 = n ¯ X n - σn 1 / 2 = n ( ¯ X n - μ ) So sometimes, the expression n ( ¯ X n - μ ) is used instead of ( S n - ) / ( σn 1 / 2 ). Date : September 24, 2004 - September 30, 2004 . 1
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2 ORIE360/560, FALL 2004 CHONG WANG Example 1 : The fracture strength of a certain type of glass average 14 (thousands of pounds per square inch) and have a standard deviation of 2. What is the probability that the average fracture strength for 100 pieces of this glass exceeds 14.5? Assume that the characteristics of the 100 pieces in the sample are independent. Solution : Here μ = 14 and σ = 2, and one needs to compute the probability P ( ¯ X 100 > 14 . 5). Use the CLT approximation
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This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.

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Section9Notes - SECTION NOTE 9 ORIE360/560 FALL 2004 CHONG...

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