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Lecture11

# Lecture11 - Central Limit Theorem(contd Central Limit...

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Central Limit Theorem (cont’d) 11/03/2005

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Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b ( n, p, j ) we have lim n →∞ npq b ( n, p, np + x npq ) = φ ( x ) , where φ ( x ) is the standard normal density. 1
Recall: The standardized sum S * n = S n - np npq . Then P ( a S n b ) = P a - np npq S * n b - np npq . 2

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Central Limit Theorem for Bernoulli Trials Theorem. Let S n be the number of successes in n Bernoulli trials with probability p for success, and let a and b be two fixed real numbers. Define a * = a - np npq and b * = b - np npq . Then lim n →∞ P ( a S n b ) = Z b * a * φ ( x ) dx . 3
How to use this theorem? The integral on the right side of this equation is equal to the area under the graph of the standard normal density φ ( x ) between a and b . We denote this area by NA ( a * , b * ) . Unfortunately, there is no simple way to integrate the function e - x 2 / 2 . 4

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NA (0,z) = area of shaded region 0 z z NA(z) z NA(z) z NA(z) z NA(z) .0 .0000 1.0 .3413 2.0 .4772 3.0 .4987 .1 .0398 1.1 .3643 2.1 .4821 3.1 .4990 .2 .0793 1.2 .3849 2.2 .4861 3.2 .4993 .3 .1179 1.3 .4032 2.3 .4893 3.3 .4995 .4 .1554 1.4 .4192 2.4 .4918 3.4 .4997 .5 .1915 1.5 .4332 2.5 .4938 3.5 .4998 .6 .2257 1.6 .4452 2.6 .4953 3.6 .4998 .7 .2580 1.7 .4554 2.7 .4965 3.7 .4999 .8 .2881 1.8 .4641 2.8 .4974 3.8 .4999 .9 .3159 1.9 .4713 2.9 .4981 3.9 .5000 5
Approximation of Binomial Probabilities Suppose that S n is binomially distributed with parameters n and p . P ( i S n j ) NA ˆ i - 1 2 - np npq , j + 1 2 - np npq ! . 6

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Example A coin is tossed 100 times. Estimate the probability that the number of heads lies between 40 and 60. 7
Example A coin is tossed 100 times. Estimate the probability that the number of heads lies between 40 and 60. The expected number of heads is 100 · 1 / 2 = 50 , and the standard deviation for the number of heads is p 100 · 1 / 2 · 1 / 2 = 5 ; n = 100 is reasonably large 7

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P (40 S n 60) P 39 . 5 - 50 5 S * n 60 . 5 - 50 5 = P ( - 2 . 1 S * n 2 . 1) NA ( - 2 . 1 , 2 . 1) = 2 NA (0 , 2 . 1) . 9642 .
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Lecture11 - Central Limit Theorem(contd Central Limit...

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