“Central Limit Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Central Limit Theorem

释义

Central Limit Theorem

Let be a set of Independent random variates and each have an arbitrary probability distribution with Mean and a finite Variance. Then the normal form variate

(1)

has a limiting distribution which is Normal (Gaussian)with Mean

and Variance

. If conversion to normal form is not performed, then thevariate

(2)

is Normally Distributed with

and

. To provethis, consider the Inverse Fourier Transform of




			(3)

Now write



	(4)

so we have









	(5)

Now expand

(6)



			(7)

since

			(8)
			(9)

Taking the Fourier Transform,


			(10)

This is of the form

(11)

where

and

. But, fromAbramowitz and Stegun (1972, p. 302, equation 7.4.6),

(12)

Therefore,



			(13)

But

and

, so

(14)

The ``fuzzy'' central limit theorem says that data which are influenced by many small and unrelated random effects areapproximately Normally Distributed.

See also Lindeberg Condition, Lindeberg-Feller Central Limit Theorem, Lyapunov Condition

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 112-113, 1992.

Zabell, S. L. ``Alan Turing and the Central Limit Theorem.'' Amer. Math. Monthly 102, 483-494, 1995.

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