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de Moivre-Laplace Theorem


The asymptotic form of the n-step Bernoulli distribution with parameters p and q=1-p is given by

P_n(k)=(n; k)p^kq^(n-k)
(1)
∼1/(sqrt(2pinpq))e^(-(k-np)^2/(2npq))
(2)

(Papoulis 1984, p. 105).

Uspensky (1937) defines the de Moivre-Laplace theorem as the fact that the sum of those terms of the binomial series of (p+q)^n for which the number of successes x falls between d_1 and d_2 is approximately

 Q approx 1/(sqrt(2pi))int_(t_1)^(t_2)e^(-t^2/2)dt,
(3)

where

t_1=(d_1-1/2-np)/sigma
(4)
t_2=(d_2+1/2-np)/sigma
(5)
sigma=sqrt(npq).
(6)

More specifically, Uspensky (1937, p. 129) showed that

 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,
(7)

where the error term satisfies

 |Omega|<(0.13+0.18|p-q|)/(sigma^2)+e^(-3sigma/2)
(8)

for sigma>=5 (Uspensky 1937, p. 129; Kenney and Keeping 1951, pp. 36-37). Note that Kenney and Keeping (1951, p. 37) give the slightly smaller denominator 0.12+0.18|p-q|.

A corollary states that the probability that x successes in n trials will differ from the expected value np by more than d is Pdelta=1-Q_delta, where

 Q_delta=2/(sqrt(2pi))int_0^deltae^(-t^2/2)dt,
(9)

with

 delta=(d+1/2)/sigma
(10)

(Kenney and Keeping 1951, p. 39). Uspensky (1937, p. 130) showed that Q_(delta_1)=P(|x-np|<=d) is given by

 Q_(delta_1)=2/(sqrt(2pi))int_0^(delta_1)e^(-u^2/2)du+(1-theta_1-theta_2)/(sqrt(2pi)sigma)e^(-delta_1^2/2)+Omega_1,
(11)

where

delta_1=d/delta
(12)
theta_1=(nq+d)-|_nq+d_|
(13)
theta_2=(np+d)-|_np+d_|,
(14)

and the error term satisfies

 |Omega_1|<(0.20+0.25|p-q|)/(sigma^2)+e^(-3sigma/2),
(15)

for sigma>=5 (Uspensky 1937, p. 130; Kenney and Keeping 1951, pp. 40-41).


See also

Bernoulli Distribution, Binomial Series, Normal Distribution, Weak Law of Large Numbers

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References

de la Vallée-Poussin, C. "Demonstration nouvelle du théorème de Bernoulli." Ann. Soc. Sci. Bruxelles 31, 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 Helvetici 2, 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.

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de Moivre-Laplace Theorem

Cite this as:

Weisstein, Eric W. "de Moivre-Laplace Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/deMoivre-LaplaceTheorem.html

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