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Classic Problems of Probability by Prakash Gorroochurn

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Problem 10

De Moivre, Gauss, and the Normal Curve (1730, 1809)

Problem. Consider the binomial expansion of img, where n is even and large. Let M be the middle term and Q be the term at a distance d from M. Show that

(10.1) equation

Solution. We let img, where m is a positive integer. Then

img

Applying Stirling's formula imgfor large N, we have

img

Taking logarithms on both sides, with d/m small, we have

img

Writing img, we have img, as required.

10.1 Discussion

The above result was first obtained by Abraham de Moivre (1667–1754) (Fig. 10.3) in an attempt to approximate the symmetric binomial distribution. De Moivre started this ...

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