Moivre-Laplace's theorem

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As the number of points increases, the discrete binomial distribution approaches the continuous normal distribution.

The Moivre-Laplace theorem , also called de Moivre-Laplace's theorem or de Moivre-Laplace's central limit theorem , is a proposition from probability theory . According to this theorem , the binomial distribution for and probabilities converges to the normal distribution . With a large sample size, the normal distribution can therefore be used as an approximation of the binomial distribution, which is used in particular for the normal approximation and for hypothesis tests . For this approximation can be illustrated experimentally by the Galton board .

From a historical point of view, Moivre-Laplace's theorem is the first central limit theorem . In 1730 Abraham de Moivre showed the testimony for and in 1812 the general case was shown by Pierre-Simon Laplace .


Let be a sequence of independent Bernoulli-distributed random variables with the parameters and . Then the sum is binomially distributed with parameters , and we have:


(2) for everyone with .

Moivre-Laplace's theorem states that the distribution of the random variables for weak converges to the normal distribution with the variance .


Moivre-Laplace's theorem is the theoretical basis of normal approximation , a method with which the binomial distribution can be approximated.

The above statement is reformulated by a substitution and the standard normal distribution is obtained with the distribution function

for everyone .

In this way, the value of the binomially distributed random variable can be approximated to the standard normal distribution via the values ​​of the distribution function. This is usually taken from the table of the standard normal distribution .

Moivre-Laplace's theorem gives sufficiently good approximations if and satisfy the following condition:

In normal approximation, a so-called continuity correction is also introduced to reduce the approximation error , which consists of the introduction of correction terms and is intended to compensate for the transition from a discrete to a continuous probability distribution.


Plot of the density of the normal distribution with μ = 12 and σ = 3 and the binomial distribution with n = 48 and p = 1/4

The following calculations are carried out to illustrate the importance of the error correction.

Given a binomial distribution with and , consequently holds . We compare with a normal distribution with a mean and a variance .

Now we are looking for the answer to the question “What is the probability that values ​​are less than or equal ”. The calculations or estimates give the following results:

  • Binomial distribution
The approximate value was taken from the adjacent plot.
  • Normal distribution with continuity correction
With this calculation it should be noted that for reasons of symmetry applies and is for .
  • Normal distribution without continuity correction

Overall, it can be deduced from the values ​​of the "calculations" that with the help of the continuity correction a better correspondence with the value of the binomial distribution is achieved.

See also


Web links

Individual evidence

  1. Norbert Henze: Stochastics for beginners. An introduction to the fascinating world of chance . 10th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03076-6 , p. 223 , doi : 10.1007 / 978-3-658-03077-3 .
  2. ^ AV Prokhorov: Laplace theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
  3. ^ Michael Sachs: Probability calculation and statistics for engineering students at technical colleges. Fachbuchverlag Leipzig, Munich 2003, ISBN 3-446-22202-2 , pp. 129-130