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[阅读笔记] 神奇的正态分布与中心极限定理

已有 6933 次阅读 2021-7-25 16:04 |个人分类:基础数学-逻辑-物理|系统分类:科研笔记

[阅读笔记] 神奇的正态分布与中心极限定理

近来，俺的[重大困惑]是为什么正态分布随机数不能“被”预测。

http://blog.sciencenet.cn/blog-107667-1296529.html 。下面是直观的叙述。不知道怎样专业地描述。主要是阅读笔记或抄录。感谢所引用的所有网站！

Gaussian distribution. The Gaussian (normal) distribution was historically calle.png

Gaussian distribution.

The Gaussian (normal) distribution was historically called the law of errors. 正态分布 normal distribution，也叫高斯分布 Gaussian distribution；历史上也叫误差定律 law of errors。

As the figure above illustrates, 68% of the values lie within 1 standard deviation of the mean; 95% lie within 2 standard deviations; and 99.7% lie within 3 standard deviations. 如上图所示，68% 的值在平均值的 1 个标准偏差内； 95% 位于 2 个标准差以内； 99.7% 位于 3 个标准偏差内。 https://introcs.cs.princeton.edu/java/11gaussian/

一、正态分布的成因

（1）一般来说，如果一个量是由许多微小的独立随机因素影响的结果，那么就可以认为这个量具有正态分布（见中心极限定理）。

https://baike.baidu.com/item/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83

（2）the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution.

具有有限均值和方差的任何分布的任何一组变量的均值趋于正态分布。

As Lippmann stated, "Everybody believes in the exponential law of errors: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation" (Whittaker and Robinson 1967, p. 179).

正如李普曼所说，“每个人都相信指数的误差定律：实验者，因为他们认为它可以被数学证明；而数学家，因为他们相信它是通过观察建立的”（惠特克和罗宾逊，1967 年，第 179 页））。

https://mathworld.wolfram.com/NormalDistribution.html

（3）In science and engineering, it is often reasonable to treat the error of an observation as the result of many small, independent, errors. This enables us to apply the central limit theorem and treat the errors as normal.

Caveat: central limit theorem typically only applies when close to the peak; may not apply in the tails.

The French physicist Gabriel Lippman wrote the following in a letter to Henri Poincare.

Tout le monde y croit cependent, car les experimenteurs s'imaginent que c'est un theorem de mathematiques, et les mathematiciens que c'est un fait experimental.

Everybody believes in the exponential law of errors: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation.

在科学和工程中，将观测误差视为许多独立的小误差的结果通常是合理的。这使我们能够应用中心极限定理并将错误视为正常。

警告：中心极限定理通常仅在接近峰值时适用；可能不适用于尾部。

法国物理学家加布里埃尔·李普曼在给亨利·庞加莱的一封信中写道。

Tout le monde y croit cependent, car les Experimentaleurs s'imaginent que c'est un theorem de mathematiques, et les mathematiciens que c'est un fait Experimental。

每个人都相信指数的误差定律：实验者，因为他们认为它可以被数学证明；而数学家，因为他们相信它是通过观察建立的。

https://introcs.cs.princeton.edu/java/11gaussian/

Gaussian Distribution - Princeton University

二、正态分布的性质

（1）Lévy-Cramér theorem:

If the sum of two independent non-constant random variables is normally distributed, then each of the summands is normally distributed.

如果两个独立的非常数随机变量的和是正态分布的，那么每个被加数都是正态分布的。

https://encyclopediaofmath.org/wiki/L%C3%A9vy-Cram%C3%A9r_theorem

（2）Gaussian and Uniform White Noise:

A white noise signal (process) is constituted by a set of independent and identically distributed (i.i.d) random variables. In discrete sense, the white noise signal constitutes a series of samples that are independent and generated from the same probability distribution. For example, you can generate a white noise signal using a random number generator in which all the samples follow a given Gaussian distribution. This is called White Gaussian Noise (WGN) or Gaussian White Noise. Similarly, a white noise signal generated from a Uniform distribution is called Uniform White Noise.

高斯和均匀白噪声：

白噪声信号（过程）由一组独立的同分布（i.i.d）随机变量构成。在离散意义上，白噪声信号构成了一系列独立的样本，这些样本是从相同的概率分布中生成的。例如，您可以使用随机数生成器生成白噪声信号，其中所有样本都遵循给定的高斯分布。这称为高斯白噪声 (WGN) 或高斯白噪声。类似地，由均匀分布产生的白噪声信号称为均匀白噪声。

https://www.gaussianwaves.com/2013/11/simulation-and-analysis-of-white-noise-in-matlab/

三、中心极限定理

（1）a number of limit theorems in probability theory stating conditions under which sums or other functions of a large number of independent or weakly-dependent random variables have a probability distribution close to the normal distribution.

概率论中的许多极限定理说：在某些条件下，大量独立的或弱相关随机变量的总和或其它函数，是一个接近正态分布的概率分布。

Numerous versions are known of generalizations of the central limit theorem to sums of dependent variables. (In the case of homogeneous finite Markov chains, the simplest non-homogeneous chains with two states, and certain other schemes; this was done by Markov himself in 1907–1911, subsequent generalizations are connected in the first instance with the name of S.N. Bernshtein [B].) A basic feature peculiar to all generalizations of this kind of the central limit theorem (if one is concerned with a triangular array) consists in the fact that the dependence between the events determined by X₁...X_k, and those determined by X_k+p, X_k+p+1… becomes vanishingly small when p grows indefinitely.

中心极限定理对因变量和的推广是已知的许多版本。（在齐次有限马尔可夫链的情况下，最简单的具有两个状态的非齐次链和某些其他方案；这是由马尔可夫自己在 1907-1911 年完成的，随后的概括首先与 S.N. Bernshtein 的名称联系起来 [B].) 这种中心极限定理（如果涉及三角形阵列）的所有推广所特有的基本特征在于，由 X₁...X_k 确定的事件之间的相关性当 p 无限增长时，X_k+p, X_k+p+1…变得非常小。

Concerning the central limit theorem in number theory see Number theory, probabilistic methods in. The central limit theorem is also applicable in certain problems in function theory and in the theory of dynamical systems.

关于数论中的中心极限定理，请参阅数论，概率方法。中心极限定理也适用于函数论和动力系统理论中的某些问题。

https://encyclopediaofmath.org/wiki/Central_limit_theorem

（2）The "fuzzy" central limit theorem says that data which are influenced by many small and unrelated random effects are approximately normally distributed.

“模糊”中心极限定理说，受许多小的和不相关的随机效应影响的数据近似正态分布。

https://mathworld.wolfram.com/CentralLimitTheorem.html

（3）有些概率分布，自变量取值都是正的实数。

如 Weibull, exponential, logarithmic, log-normal, von Mises, Rayleigh, Erlang, Gamma, Maxwell–Boltzmann, 等。

根据Central limit theorem，当 sums or other functions of a large number of independent or weakly-dependent random variables have a probability distribution close to the normal distribution.

即当求和的项数越来越大时，这些“自变量取正实数”的分布求和也趋向于正态分布。

http://blog.sciencenet.cn/blog-107667-1296695.html

感谢 Encyclopedia of Mathematics 和 Wolfram 等网站的重要资料！

祝福您们！

参考资料：

[1] Normal distribution. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Normal_distribution

[2] Central limit theorem. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Central_limit_theorem

[3] Weisstein, Eric W. "Normal Distribution." From MathWorld--A Wolfram Web Resource

https://mathworld.wolfram.com/NormalDistribution.html

[4] Wolfram Research (2007)，NormalDistribution，Wolfram 语言函数，（更新于 2016 ）.

https://reference.wolfram.com/language/ref/NormalDistribution.html

[5] Weisstein, Eric W. "Central Limit Theorem." From MathWorld--A Wolfram Web Resource.

https://mathworld.wolfram.com/CentralLimitTheorem.html

The "fuzzy" central limit theorem says that data which are influenced by many small and unrelated random effects are approximately normally distributed.

[6] Lévy-Cramér theorem. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/L%C3%A9vy-Cram%C3%A9r_theorem

Theorems analogous to the Lévy–Cramér theorem have been obtained for the Poisson distribution (Raikov's theorem), for the convolution of a Poisson and a normal distribution, and for other classes of infinitely-divisible distributions (see [6]).

[7] Fourier transform. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Fourier_transform

[8] Spectral density. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Spectral_density

[9] Errors, theory of. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/Errors,_theory_of

[10] White noise. Encyclopedia of Mathematics.

https://encyclopediaofmath.org/wiki/White_noise

[11] GaussianWaves, 2013-11-29, White Noise : Simulation and Analysis using Matlab

https://www.gaussianwaves.com/2013/11/simulation-and-analysis-of-white-noise-in-matlab/

[12] 正态分布 - 百度百科

https://baike.baidu.com/item/%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83