# Laplace theorem

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Laplace's theorem on determinants. See Cofactor.

Laplace's theorem on the approximation of the binomial distribution by the normal distribution. This is the first version of the central limit theorem of probability theory: If is the number of "successes" in Bernoulli trials with probability of success , , then, as , for any real numbers and () one has

 (*)

where

is the distribution function of the standard normal law.

The local Laplace theorem has independent significance: For the probability

one has

where

is the density of the standard normal distribution and as uniformly for all for which belongs to some finite interval.

In its general form the theorem was proved by P.S. Laplace [1]. The special case of the Laplace theorem was studied by A. de Moivre [2], and therefore the Laplace theorem is sometimes called the de Moivre–Laplace theorem.

For practical applications the Laplace theorem is important in order to obtain an idea of the errors that arise in the use of approximate formulas. In the more precise (by comparison with [1]) asymptotic formula

the remainder term has order uniformly for all real . For uniform approximation of the binomial distribution by means of the normal distribution the formula of Ya. Uspenskii (1937) is more useful: If , then for any and ,

where

and for ,

To improve the relative accuracy of the approximation S.N. Bernstein [S.N. Bernshtein] (1943) and W. Feller (1945) suggested other formulas.

#### References

 [1] P.S. Laplace, "Théorie analytique des probabilités" , Paris (1812) [2] A. de Moivre, "Miscellanea analytica de seriebus et quadraturis" , London (1730) [3] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) [4] W. Feller, "On the normal approximation to the binomial distribution" Ann. Math. Statist. , 16 (1945) pp. 319–329 [5] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968)