Problem 10

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

Problem. Consider the binomial expansion of , 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)

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

Applying Stirling's formula for large N, we have

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

Writing , we have , 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 ...