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
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.
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 ...