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is the density of the standard normal distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753015.png" /> uniformly for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753017.png" /> belongs to some finite interval.
 
is the density of the standard normal distribution and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753015.png" /> uniformly for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753017.png" /> belongs to some finite interval.
  
In its general form the theorem was proved by P.S. Laplace [[#References|[1]]]. The special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753018.png" /> of the Laplace theorem was studied by A. de Moivre [[#References|[2]]], and therefore the Laplace theorem is sometimes called the de Moivre–Laplace theorem.
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In its general form the theorem was proved by P.S. Laplace {{Cite|L}}. The special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753018.png" /> of the Laplace theorem was studied by A. de Moivre {{Cite|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 [[#References|[1]]]) asymptotic formula
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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 {{Cite|L}}) asymptotic formula
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753019.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057530/l05753019.png" /></td> </tr></table>
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Laplace, "Théorie analytique des probabilités" , Paris (1812) {{MR|2274728}} {{MR|1400403}} {{MR|1400402}} {{ZBL|1047.01534}} {{ZBL|1047.01533}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. de Moivre, "Miscellanea analytica de seriebus et quadraturis" , London (1730)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W. Feller, "On the normal approximation to the binomial distribution" ''Ann. Math. Statist.'' , '''16''' (1945) pp. 319–329 {{MR|0015706}} {{ZBL|0060.28703}} </TD></TR>
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{|
<TR><TD valign="top">[5]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1968) </TD></TR></table>
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|valign="top"|{{Ref|L}}|| P.S. Laplace, "Théorie analytique des probabilités" , Paris (1812) {{MR|2274728}} {{MR|1400403}} {{MR|1400402}} {{ZBL|1047.01534}} {{ZBL|1047.01533}}
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|-
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|valign="top"|{{Ref|M}}|| A. de Moivre, "Miscellanea analytica de seriebus et quadraturis" , London (1730)
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|-
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|valign="top"|{{Ref|PR}}|| Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}}
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|-
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|valign="top"|{{Ref|F}}|| W. Feller, "On the normal approximation to the binomial distribution" ''Ann. Math. Statist.'' , '''16''' (1945) pp. 319–329 {{MR|0015706}} {{ZBL|0060.28703}}
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|-
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|valign="top"|{{Ref|F2}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1968)
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|}
  
 
====Comments====
 
====Comments====
For a more detailed and more general discussion of approximation by the [[Normal distribution|normal distribution]] see [[#References|[a1]]].
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For a more detailed and more general discussion of approximation by the [[Normal distribution|normal distribution]] see {{Cite|P}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}} </TD></TR></table>
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{|
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|valign="top"|{{Ref|P}}|| V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
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|}

Revision as of 17:41, 13 May 2012

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

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 [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

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

[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
How to Cite This Entry:
Laplace theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laplace_theorem&oldid=25540
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article