Moivre-Laplace's theorem
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 .
statement
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:
(1)
(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 .
Applications
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.
example
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
literature
- 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 , doi: 10.1007 / 978-3-658-03077-3 , p. 221 ff.
- Ulrich Krengel : Introduction to probability theory and statistics. 7th edition. Vieweg-Verlag, Wiesbaden 2003, ISBN 3-528-67259-5 , doi: 10.1007 / 978-3-322-93581-6 , pp. 80-83
Web links
- Binomial and normal distribution - online tutorial with dynamic worksheets (Java plugin required)
Individual evidence
- ↑ 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 .
- ^ AV Prokhorov: Laplace theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- ^ Michael Sachs: Probability calculation and statistics for engineering students at technical colleges. Fachbuchverlag Leipzig, Munich 2003, ISBN 3-446-22202-2 , pp. 129-130