PROBABILITY THEORY:CENTRAL LIMIT THEOREM

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CENTRAL LIMIT THEOREM

Section 24.6 pointed out that sums of independent normally distributed random variables are themselves normally distributed, and that even if the random variables are not normally distributed, the distribution of their sum still tends toward normality. This latter statement can be made precise by means of the central limit theorem.

Central Limit Theorem

Let the random variables X1, X2, . . . , Xn be independent with means JL1, JL2, . . . , JLn, re- spectively, and variance u2, u2, . . . , u2, respectively. Consider the random variable Z ,

INTRODUCTION TO OPERATIONS RESEARCH-0593

Note that if the Xi form a random sample, with each Xi having mean JL and variance u2, then Zn = (X - JL)Vn /u.† Hence, sample means from random samples tend toward nor- mality in the sense just described by the central limit theorem even if the Xi are not normally distributed.

It is difficult to give sample sizes beyond which the central limit theorem applies and approximate normality can be assumed for sample means. This, of course, does depend upon the form of the underlying distribution. From a practical point of view, moderate sample sizes, like 10, are often sufficient.

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