Moivre-Laplace's theorem

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 . ${\displaystyle n\rightarrow \infty }$ ${\displaystyle 0<p<1}$ ${\displaystyle p={\tfrac {1}{2}}}$

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 . ${\displaystyle p={\tfrac {1}{2}}}$

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: ${\displaystyle X_{1},X_{2},X_{3},\cdots }$ ${\displaystyle p\in \;]0,1[}$ ${\displaystyle \sigma ^{2}=p(1-p)}$ ${\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle p\in \;]0,1[}$

(1) ${\displaystyle \quad \operatorname {P} \left(S_{n}=k\right)\;=\;B(k\mid p,n)\;\approx \;{\frac {1}{\sqrt {2\pi n\sigma ^{2}}}}\,\exp \left(-{\frac {n}{2\sigma ^{2}}}\left({\frac {k}{n}}-p\right)^{2}\right)}$

(2) for everyone with . ${\displaystyle \quad \lim _{n\to \infty }\operatorname {P} \left(x_{1}\leq {\frac {S_{n}-np}{\sqrt {n}}}\leq x_{2}\right)\;={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\int _{x_{1}}^{x_{2}}\exp \left(-{x^{2} \over 2\sigma ^{2}}\right)\mathrm {d} x}$ ${\displaystyle x_{1},x_{2}\in \mathbb {R} }$ ${\displaystyle x_{1}<x_{2}}$

Moivre-Laplace's theorem states that the distribution of the random variables for weak converges to the normal distribution with the variance . ${\displaystyle \textstyle {\frac {S_{n}-np}{\sqrt {n}}}}$ ${\displaystyle n\to \infty }$ ${\displaystyle {\mathcal {N}}\left(0,\sigma ^{2}\right)}$ ${\displaystyle \sigma ^{2}=p(1-p)}$

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 ${\displaystyle \Phi }$

${\displaystyle \lim _{n\to \infty }\left|\operatorname {P} (S_{n}\leq t)-\Phi \left({\frac {t-np}{\sqrt {np(1-p)}}}\right)\right|=0}$

for everyone . ${\displaystyle t\in \mathbb {R} }$

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 . ${\displaystyle S_{n}}$

Moivre-Laplace's theorem gives sufficiently good approximations if and satisfy the following condition: ${\displaystyle n}$ ${\displaystyle p}$

${\displaystyle np(1-p)>9}$

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. ${\displaystyle \pm {\tfrac {1}{2}}}$

example

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 . ${\displaystyle n=48}$ ${\displaystyle p={\tfrac {1}{4}}}$ ${\displaystyle np(1-p)=9}$ ${\displaystyle \mu =np=12}$ ${\displaystyle \sigma ^{2}=np(1-p)=9}$

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: ${\displaystyle \mu -3\sigma =3}$

• Binomial distribution

${\displaystyle P(0\leq X\leq 3)=\sum _{k=0}^{3}{\binom {48}{k}}\cdot {\bigg (}{\frac {1}{4}}{\bigg )}^{k}\cdot {\bigg (}{\frac {3}{4}}{\bigg )}^{48-k}\approx 0{,}004}$

The approximate value was taken from the adjacent plot.

• Normal distribution with continuity correction

${\displaystyle {\begin{aligned}P(0\leq X\leq 3)&\approx \Phi \left({\tfrac {3+0{,}5-12}{3}}\right)-\Phi \left({\tfrac {0-0{,}5-12}{3}}\right)=\Phi \left(-2{,}83\right)-\Phi \left(-4{,}17\right)\\&=1-\Phi \left(2{,}83\right)-1+\Phi \left(4{,}17\right)\approx -0{,}99767+1=0{,}00233\end{aligned}}}$

With this calculation it should be noted that for reasons of symmetry applies and is for . ${\displaystyle \Phi (-x)=1-\Phi (x)}$ ${\displaystyle \Phi (x)\approx 1}$ ${\displaystyle x>4{,}09}$

• Normal distribution without continuity correction

${\displaystyle {\begin{aligned}P(0\leq X\leq 3)&\approx \Phi \left({\tfrac {3-12}{3}}\right)-\Phi \left({\tfrac {0-12}{3}}\right)=\Phi \left(-3\right)-\Phi \left(-4\right)\\&=1-\Phi \left(3\right)-1+\Phi \left(4\right)=-0{,}99865+0{,}99997=0{,}00132\end{aligned}}}$

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

• Central limit theorem

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

Wikibooks: Proof of Moivre-Laplace's Theorem

• Binomial and normal distribution - online tutorial with dynamic worksheets (Java plugin required)

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