概率论与数理统计英文文献

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Introduction to probability theory and

mathematical statistics

The theory of probability and the mathematical statistic are carries on deductive and the induction science to the stochastic phenomenon statistical rule, from the quantity side research stochastic phenomenon statistical regular foundation mathematics discipline, the theory of probability and the mathematical statistic may divide into the theory of probability and the mathematical statistic two branches. The probability uses for the possible size quantity which portrays the random event to occur. Theory of probability main content including classical generally computation, random variable distribution and characteristic numeral and limit theorem and so on. The mathematical statistic is one of mathematics Zhonglian department actually most directly most widespread branches, it introduced an estimate (rectangular method estimate, enormousestimate), the parameter supposition examination, the non-parameter supposition examination, the variance analysis and the multiple regression analysis, the fail-safe analysis and so on the elementary knowledge and the principle, enable the student to have a profound understanding to statistics principle function. Through this curriculum study, enables the student comprehensively to understand, to grasp the theory of probability and the mathematical statistic thought and the method, grasps basic and the commonly used analysis and the computational method, and can studies in the solution economy and the management practice question using the theory of probability and the mathematical statistic viewpoint and the method. Random phenomenon

From random phenomenon, in the nature and real life, some things are interrelated and continuous development. In the relationship between each other and developing, according to whether there is a causal relationship, very different can be divided into two categories: one is deterministic phenomenon. This kind of phenomenon is under certain conditions, will lead to certain results. For example, under normal atmospheric pressure,

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water heated to 100 degrees Celsius, is bound to a boil. This link is belong to the inevitability between things. Usually in natural science is interdisciplinary studies and know the inevitability, seeking this kind of inevitable phenomenon.Another kind is the phenomenon of uncertainty. This kind of phenomenon is under certain conditions, the result is uncertain. The same workers on the same machine tools, for example, processing a number of the same kind of parts, they are the size of the there will always be a little difference. As another example, under the same conditions, artificial accelerating germination test of wheat varieties, each tree seed germination is also different, there is strength and sooner or later, respectively, and so on. Why in the same situation, will appear this kind of uncertain results? This is because, we say "same conditions" refers to some of the main conditions, in addition to these main conditions, there are many minor conditions and the accidental factor is people can't in advance one by one to grasp. Because of this, in this kind of phenomenon, we can't use the inevitability of cause and effect, the results of individual phenomenon in advance to make sure of the answer. The relationship between things is belong to accidental, this phenomenon is called accidental phenomenon, or a random phenomenon.

In nature, in the production, life, random phenomenon is very common, that is to say, there is a lot of random phenomenon. Issue such as: sports lottery of the winning Numbers, the same production line production, the life of the bulb, etc., is a random phenomenon. So we say: random phenomenon is: under the same conditions, many times the same test or survey the same phenomenon, the results are not identical, and unable to accurately predict the results of the next. Random phenomena in the uncertainties of the results, it is because of some minor, caused by the accidental factors.

Random phenomenon on the surface, seems to be messy, there is no regular phenomenon. But practice has proved that if the same kind of a large number of repeated random phenomenon, its overall present certain regularity. A large number of similar random phenomena of this kind of regularity, as we observed increase in the number of the number of times and more obvious. Flip a coin, for example, each throw is difficult to judge on that side, but if repeated many times of toss the coin, it will be more and more

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clearly find them up is approximately the same number.

We call this presented by a large number of similar random phenomena of collective regularity, is called the statistical regularity. Probability theory and mathematical statistics is the study of a large number of similar random phenomena statistical regularity of the mathematical disciplines.

The emergence and development of probability theory

Probability theory was created in the 17th century, it is by the development of insurance business, but from the gambler's request, is that mathematicians thought the source of problem in probability theory.

As early as in 1654, there was a gambler may tired to the mathematician PASCAL proposes a question troubling him for a long time: "meet two gamblers betting on a number of bureau, who will win the first m innings wins, all bets will be who. But when one of them wins a (a < m), the other won b (b < m) bureau, gambling aborted. Q: how should bets points method is only reasonable?" Who in 1642 invented the world's first mechanical addition of computer.

Three years later, in 1657, the Dutch famous astronomy, physics, and a mathematician huygens is trying to solve this problem, the results into a book concerning the calculation of a game of chance, this is the earliest probability theory works.

In recent decades, with the vigorous development of science and technology, the application of probability theory to the national economy, industrial and agricultural production and interdisciplinary field. Many of applied mathematics, such as information theory, game theory, queuing theory, cybernetics, etc., are based on the theory of probability.

Probability theory and mathematical statistics is a branch of mathematics, random they similar disciplines are closely linked. But should point out that the theory of probability and mathematical statistics, statistical methods are each have their own contain different content.

Probability theory, is based on a large number of similar random phenomena

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statistical regularity, the possibility that a result of random phenomenon to make an objective and scientific judgment, the possibility of its occurrence for this size to make quantitative description; Compare the size of these possibilities, study the contact between them, thus forming a set of mathematical theories and methods.

Mathematical statistics - is the application of probability theory to study the phenomenon of large number of random regularity; To through the scientific arrangement of a number of experiments, the statistical method given strict theoretical proof; And determining various methods applied conditions and reliability of the method, the formula, the conclusion and limitations. We can from a set of samples to decide whether can with quite large probability to ensure that a judgment is correct, and can control the probability of error.

- is a statistical method provides methods are used in a variety of specific issues, it does not pay attention to the method according to the theory, mathematical reasoning.

Should point out that the probability and statistics on the research method has its particularity, and other mathematical subject of the main differences are:

First, because the random phenomena statistical regularity is a collective rule, must to present in a large number of similar random phenomena, therefore, observation, experiment, research is the cornerstone of the subject research methods of probability and statistics. But, as a branch of mathematics, it still has the definition of this discipline, axioms, theorems, the definitions and axioms, theorems are derived from the random rule of nature, but these definitions and axioms, theorems is certain, there is no randomness.

Second, in the study of probability statistics, using the "by part concluded all" methods of statistical inference. This is because it the object of the research - the range of random phenomenon is very big, at the time of experiment, observation, not all may be unnecessary. But by this part of the data obtained from some conclusions, concluded that the reliability of the conclusion to all the scope.

Third, the randomness of the random phenomenon, refers to the experiment, investigation before speaking. After the real results for each test, it can only get the results

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of the uncertainty of a certain result. When we study this phenomenon, it should be noted before the test can find itself inherent law of this phenomenon.

The content of the theory of probability

Probability theory as a branch of mathematics, it studies the content general include the probability of random events, the regularity of statistical independence and deeper administrative levels.

Probability is a quantitative index of the possibility of random events. In independent random events, if an event frequency in all events, in a larger range of stable around a fixed constant. You can think the probability of the incident to the constant. For any event probability value must be between 0 and 1.

There is a certain type of random events, it has two characteristics: first, only a finite number of possible results; Second, the results the possibility of the same. Have the characteristics of the two random phenomenon called "classical subscheme".

In the objective world, there are a large number of random phenomena, the result of a random phenomenon poses a random event. If the variable is used to describe each random phenomenon as a result, is known as random variables.

Random variable has a finite and the infinite, and according to the variable values is usually divided into discrete random variables and the discrete random variable. List all possible values can be according to certain order, such a random variable is called a discrete random variable; If possible values with an interval, unable to make the order list, the random variable is called a discrete random variable.

The content of the mathematical statistics

Including sampling, optimum line problem of mathematical statistics, hypothesis testing, analysis of variance, correlation analysis, etc. Sampling inspection is to pair through sample investigation, to infer the overall situation. Exactly how much sampling, this is a very important problem, therefore, is produced in the sampling inspection "small sample theory", this is in the case of the sample is small, the analysis judgment theory.

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Also called curve fitting and optimal line problem. Some problems need to be according to the experience data to find a theoretical distribution curve, so that the whole problem get understanding. But according to what principles and theoretical curve? How to compare out of several different curve in the same issue? Selecting good curve, is how to determine their error? ...... Is belong to the scope of the optimum line issues of mathematical statistics.

Hypothesis testing is only at the time of inspection products with mathematical statistical method, first make a hypothesis, according to the result of sampling in reliable to a certain extent, to judge the null hypothesis.

Also called deviation analysis, variance analysis is to use the concept of variance to analyze by a handful of experiment can make the judgment.

Due to the random phenomenon is abundant in human practical activities, probability and statistics with the development of modern industry and agriculture, modern science and technology and continuous development, which formed many important branch. Such as stochastic process, information theory, experimental design, limit theory, multivariate analysis, etc.

译文：

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概率论和数理统计简介

概率论与数理统计是对随机现象的统计规律进行演绎和归纳的科学，从数量侧面研究随机现象的统计规律性的基础数学学科，概率论与数理统计又可分为概率论和数理统计两个分支。概率是用来刻画随机事件发生的可能性大小的量。概率论的主要内容包括古典概型的计算、随机变量的分布及特征数字和极限定理等等。数理统计乃数学中联系实际最直接最广泛的分支之一，它介绍了点估计(矩法估计、极大似然估计)、参数假设检验、非参数假设检验、方差分析和多元回归分析、、可靠性分析等基本知识和原理，使学生对统计学原理的作用有一深刻的了解。通过本课程的学习，使学生能全面理解、掌握概率论与数理统计的思想与方法，掌握基本而常用的分析和计算方法，并能运用概率论与数理统计的观点和方法来研究解决经济与管理中的实践问题。

随机现象

从随机现象说起，在自然界和现实生活中，一些事物都是相互联系和不断 发展的。在它们彼此间的联系和发展中，根据它们是否有必然的因果联系，可以分成截然不同的两大类：一类是确定性的现象。这类现象是在一定条件下，必定会导致某种确定的结果。举例来说，在标准大气压下，水加热到100摄氏度，就必然会沸腾。事物间的这种联系是属于必然性的。通常的自然科学各学科就是专门研究和认识这种必然性的，寻求这类必然现象的因果关系，把握它们之间的数量规律。

另一类是不确定性的现象。这类现象是在一定条件下，它的结果是不确定的。举例来说，同一个工人在同一台机床上加工同一种零件若干个，它们的尺寸总会有一点差异。又如，在同样条件下，进行小麦品种的人工催芽试验，各棵种子的发芽情况也不尽相同，有强弱和早晚的分别等等。为什么在相同的情况下，会出现这种不确定的结果呢？这是因为，我们说的“相同条件”是指一些主要条件来说的，除了这些主要条件外，还会有许多次要条件和偶然因素又是人们无法事先一一能够掌握的。正因为这样，我们在这一类现象中，就无法用必然性的因果关系，对个别现象的结果事先做出确定的答案。事物间的这种关系是属于偶然性的，这种现象叫做偶然现象，或者叫做随机现象。

在自然界，在生产、生活中，随机现象十分普遍，也就是说随机现象是大量存在的。比如：每期体育彩票的中奖号码、同一条生产线上生产的灯泡的寿命等，都是随机现象。因此，我们说：随机现象就是：在同样条件下，多次进行同一试验或调查同一现象，所的结果不完全一样，而且无法准确地预测下一次所得结果的现象。随机现象这种结果的不确定性，是由于一些次要的、偶然的因素影响所造成的。

随机现象从表面上看，似乎是杂乱无章的、没有什么规律的现象。但实践证明，如果同类的随机现象大量重复出现，它的总体就呈现出一定的规律性。大量同类随机

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现象所呈现的这种规律性，随着我们观察的次数的增多而愈加明显。比如掷硬币，每一次投掷很难判断是那一面朝上，但是如果多次重复的掷这枚硬币，就会越来越清楚的发现它们朝上的次数大体相同。

我们把这种由大量同类随机现象所呈现出来的集体规律性，叫做统计规律性。概率论和数理统计就是研究大量同类随机现象的统计规律性的数学学科。

概率论的产生和发展

概率论产生于十七世纪，本来是由保险事业的发展而产生的，但是来自于赌博者的请求，却是数学家们思考概率论中问题的源泉。

早在1654年，有一个赌徒梅累向当时的数学家帕斯卡提出一个使他苦恼了很久的问题：“两个赌徒相约赌若干局，谁先赢 m局就算赢，全部赌本就归谁。但是当其中一个人赢了 a (a<m)局，另一个人赢了 b(b<m)局的时候，赌博中止。问：赌本应该如何分法才合理？”后者曾在1642年发明了世界上第一台机械加法计算机。

三年后，也就是1657年，荷兰著名的天文、物理兼数学家惠更斯企图自己解决这一问题，结果写成了《论机会游戏的计算》一书，这就是最早的概率论著作。

近几十年来，随着科技的蓬勃发展，概率论大量应用到国民经济、工农业生产及各学科领域。许多兴起的应用数学，如信息论、对策论、排队论、控制论等，都是以概率论作为基础的。

概率论和数理统计是一门随机数学分支，它们是密切联系的同类学科。但是应该指出，概率论、数理统计、统计方法又都各有它们自己所包含的不同内容。

概率论——是根据大量同类随机现象的统计规律，对随机现象出现某一结果的可能性作出一种客观的科学判断，对这种出现的可能性大小做出数量上的描述；比较这些可能性的大小、研究它们之间的联系，从而形成一整套数学理论和方法。

数理统计——是应用概率的理论来研究大量随机现象的规律性；对通过科学安排的一定数量的实验所得到的统计方法给出严格的理论证明；并判定各种方法应用的条件以及方法、公式、结论的可靠程度和局限性。使我们能从一组样本来判定是否能以相当大的概率来保证某一判断是正确的，并可以控制发生错误的概率。

统计方法——是一上提供的方法在各种具体问题中的应用，它不去注意这些方法的的理论根据、数学论证。

应该指出，概率统计在研究方法上有它的特殊性，和其它数学学科的主要不同点有：

第一，由于随机现象的统计规律是一种集体规律，必须在大量同类随机现象中才能呈现出来，所以，观察、试验、调查就是概率统计这门学科研究方法的基石。但是，

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作为数学学科的一个分支，它依然具有本学科的定义、公理、定理的，这些定义、公理、定理是来源于自然界的随机规律，但这些定义、公理、定理是确定的，不存在任何随机性。

第二，在研究概率统计中，使用的是“由部分推断全体”的统计推断方法。这是因为它研究的对象——随机现象的范围是很大的，在进行试验、观测的时候，不可能也不必要全部进行。但是由这一部分资料所得出的一些结论，要全体范围内推断这些结论的可靠性。

第三，随机现象的随机性，是指试验、调查之前来说的。而真正得出结果后，对于每一次试验，它只可能得到这些不确定结果中的某一种确定结果。我们在研究这一现象时，应当注意在试验前能不能对这一现象找出它本身的内在规律。

概率论的内容

概率论作为一门数学分支，它所研究的内容一般包括随机事件的概率、统计独立性和更深层次上的规律性。

概率是随机事件发生的可能性的数量指标。在独立随机事件中，如果某一事件在全部事件中出现的频率，在更大的范围内比较明显的稳定在某一固定常数附近。就可以认为这个事件发生的概率为这个常数。对于任何事件的概率值一定介于 0和 1之间。

有一类随机事件，它具有两个特点：第一，只有有限个可能的结果；第二，各个结果发生的可能性相同。具有这两个特点的随机现象叫做“古典概型”。

在客观世界中，存在大量的随机现象，随机现象产生的结果构成了随机事件。如果用变量来描述随机现象的各个结果，就叫做随机变量。

随机变量有有限和无限的区分，一般又根据变量的取值情况分成离散型随机变量和非离散型随机变量。一切可能的取值能够按一定次序一一列举，这样的随机变量叫做离散型随机变量；如果可能的取值充满了一个区间，无法按次序一一列举，这种随机变量就叫做非离散型随机变量。

在离散型随机变量的概率分布中，比较简单而应用广泛的是二项式分布。如果随机变量是连续的，都有一个分布曲线，实践和理论都证明：有一种特殊而常用的分布，它的分布曲线是有规律的，这就是正态分布。正态分布曲线取决于这个随机变量的一些表征数，其中最重要的是平均值和差异度。平均值也叫数学期望，差异度也就是标准方差。

数理统计的内容

数理统计包括抽样、适线问题、假设检验、方差分析、相关分析等内容。抽样检验是要通过对子样的调查，来推断总体的情况。究竟抽样多少，这是十分重要的问题，

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因此，在抽样检查中就产生了“小样理论”，这是在子样很小的情况下，进行分析判断的理论。

适线问题也叫曲线拟和。有些问题需要根据积累的经验数据来求出理论分布曲线，从而使整个问题得到了解。但根据什么原则求理论曲线？如何比较同一问题中求出的几种不同曲线？选配好曲线，有如何判断它们的误差？……就属于数理统计中的适线问题的讨论范围。

假设检验是只在用数理统计方法检验产品的时候，先作出假设，在根据抽样的结果在一定可靠程度上对原假设做出判断。

方差分析也叫做离差分析，就是用方差的概念去分析由少数试验就可以做出的判断。

由于随机现象在人类的实际活动中大量存在，概率统计随着现代工农业、近代科技的发展而不断发展，因而形成了许多重要分支。如：随机过程、信息论、极限理论、试验设计、多元分析等。