Laplace theorem
Laplace's theorem on determinants. See Cofactor.
2020 Mathematics Subject Classification: Primary: 60F05 [MSN][ZBL]
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
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is the distribution function of the standard normal law.
The local Laplace theorem has independent significance: For the probability
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one has
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where
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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 [L]. The special case of the Laplace theorem was studied by A. de Moivre [M], 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 [L]) asymptotic formula
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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
,
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where
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and for ,
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To improve the relative accuracy of the approximation S.N. Bernstein [S.N. Bernshtein] (1943) and W. Feller (1945) suggested other formulas.
References
[L] | P.S. Laplace, "Théorie analytique des probabilités" , Paris (1812) MR2274728 MR1400403 MR1400402 Zbl 1047.01534 Zbl 1047.01533 |
[M] | A. de Moivre, "Miscellanea analytica de seriebus et quadraturis" , London (1730) |
[PR] | Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754 |
[F] | W. Feller, "On the normal approximation to the binomial distribution" Ann. Math. Statist. , 16 (1945) pp. 319–329 MR0015706 Zbl 0060.28703 |
[F2] | W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1968) |
Comments
For a more detailed and more general discussion of approximation by the normal distribution see [P].
References
[P] | V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042 |
Laplace theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_theorem&oldid=39015