Uspensky (1937) defines the de Moivre-Laplace theorem as the fact that the sum of those terms of the binomial series of for which the number of successes falls between and is approximately
(3)
where
(4)
(5)
(6)
More specifically, Uspensky (1937, p. 129) showed that
(7)
where the error term satisfies
(8)
for
(Uspensky 1937, p. 129; Kenney and Keeping 1951, pp. 36-37). Note that
Kenney and Keeping (1951, p. 37) give the slightly smaller denominator.
A corollary states that the probability that successes in trials will differ from the expected value by more than is , where
(9)
with
(10)
(Kenney and Keeping 1951, p. 39). Uspensky (1937, p. 130) showed that
is given by
(11)
where
(12)
(13)
(14)
and the error term satisfies
(15)
for
(Uspensky 1937, p. 130; Kenney and Keeping 1951, pp. 40-41).
de la Vallée-Poussin, C. "Demonstration nouvelle du théorème de Bernoulli." Ann. Soc. Sci. Bruxelles31,
219-236, 1907.de Moivre, A. Miscellanea analytica. Lib. 5,
1730.de Moivre, A. The
Doctrine of Chances, or, a Method of Calculating the Probabilities of Events in Play,
3rd ed. New York: Chelsea, 2000. Reprint of 1756 3rd ed. Original ed. published
1716.Kenney, J. F. and Keeping, E. S. "The DeMoivre-Laplace
Theorem" and "Simple Sampling of Attributes." §2.10 and 2.11
in Mathematics
of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 36-41,
1951.Laplace, P. Théorie analytiques de probabilités,
3ème éd., revue et augmentée par l'auteur. Paris: Courcier,
1820. Reprinted in Œuvres complètes de Laplace, tome 7. Paris:
Gauthier-Villars, pp. 280-285, 1886.Mirimanoff, D. "Le jeu
de pile ou face et les formules de Laplace et de J. Eggenberger." Commentarii
Mathematici Helvetici2, 133-168, 1930.Papoulis, A. Probability,
Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill,
1984.Uspensky, J. V. "Approximate Evaluation of Probabilities
in Bernoullian Case." Ch. 7 in Introduction
to Mathematical Probability. New York: McGraw-Hill, pp. 119-138, 1937.
-step Bernoulli distribution
with parameters 
and 
is given by
for which the number of successes 
falls between 
and 
is approximately
![Q=1/(sqrt(2pi))int_(t_1)^(t_2)e^(-t^2/2)dt+(q-p)/(6sqrt(2pi)sigma)[(1-t^2)e^(-t^2/2)]_(t_1)^(t_2)+Omega,](/images/equations/deMoivre-LaplaceTheorem/NumberedEquation2.svg)
(Uspensky 1937, p. 129; Kenney and Keeping 1951, pp. 36-37). Note that
Kenney and Keeping (1951, p. 37) give the slightly smaller denominator 
.
successes in 
trials will differ from the expected value 
by more than 
is 
, where
is given by
(Uspensky 1937, p. 130; Kenney and Keeping 1951, pp. 40-41).
