probability theory

The branch of mathematics that studies the quantitative laws of random phenomena. Random phenomena are relative to decisive phenomena. The phenomenon that a certain result must occur under certain conditions is called decisive phenomenon. For example, at standard atmospheric pressure, when pure water is heated to 100℃, water will inevitably boil. Random phenomenon refers to the phenomenon that a series of experiments or observations will get different results under the same basic conditions. Before each experiment or observation, it is uncertain what kind of results will appear, which shows contingency. For example, flipping a coin may have a positive side or a negative side, and the life of light bulbs produced under the same technological conditions is uneven. The realization and observation of random phenomena are called random experiments. Every possible result of random test is called a basic event, and a basic event or a group of basic events is collectively called a random event, or simply called an event. The probability of an event is a measure of the possibility of an event. Although the occurrence of an event in random trials is accidental, those random trials that can be repeated in large numbers under the same conditions often show obvious quantitative laws. For example, if you throw a uniform coin several times in a row, with the increase of throwing times, the frequency of head appearance will gradually tend to 1/2. For another example, when the length of an object is measured many times, with the increase of the number of measurements, the average value of the measurement results gradually stabilizes at a constant, and most of the measured values fall near this constant, and its distribution presents a certain degree of symmetry. The law of large numbers and the central limit theorem describe and demonstrate these laws. In real life, people often need to study the evolution of a specific random phenomenon. For example, tiny particles in a liquid are randomly collided by surrounding molecules to form irregular motion (Brownian motion), which is a random process. The statistical characteristics of random process, the calculation of the probability of some events related to random process, especially the research on the sample trajectory of random process (that is, the one-time realization of the process) are the main topics of modern probability theory. Probability theory is closely related to real life and widely used in natural science, technical science, social science, military and industrial and agricultural production.

The origin of probability theory is related to gambling. /kloc-in the 6th century, Italian scholars began to study some simple problems in gambling such as dice. /kloc-In the middle of the 7th century, French mathematicians B Pascal, P de Fermat and Dutch mathematician C Huygens studied some complicated gambling problems based on permutation and combination method, and they solved the problems of dividing bets and gamblers losing money. With the development of science in 18 and 19 centuries, people noticed some similarities between some biological, physical and social phenomena and games of chance, so the probability theory originated from games of chance was applied to these fields. At the same time, it also greatly promoted the development of probability theory itself. The founder of probability theory as a branch of mathematics is Swiss mathematician J. Bernoulli, who established the first limit theorem in probability theory, namely Bernoulli's law of large numbers, and expounded the probability that the frequency of events would be stable here. Then A.de de moivre and P's Laplasse derived the original form of the second basic limit theorem (central limit theorem). On the basis of summarizing predecessors' work systematically, Laplace wrote the Theory of Probability of Analysis, gave a clear classical definition of probability, and introduced more powerful analytical tools into probability theory, pushing it to a new development stage. At the end of 19, Russian mathematicians P.L. Chebyshev, A.A. Markov, A.M. Lyapunov and others established the general forms of the law of large numbers and the central limit theorem by analytical methods, and scientifically explained why many random variables encountered in practice obey the normal distribution approximately. Stimulated by physics in the early 20th century, people began to study stochastic processes. In this regard, André Andrey Kolmogorov, Weiner, Markov, Hinchin, Levi and Ferrer all made outstanding contributions.

How to define probability and how to base probability theory on strict logic is a difficult point in the development of probability theory, and the exploration of this problem has been going on for three centuries. Lebesgue's theory of measurement and integration, which was completed in the early 20th century, and the abstract theory of measurement and integration, which was developed later, laid the foundation for the establishment of axiomatic system of probability theory. Under this background, the Soviet mathematician André Andrey Kolmogorov gave the definition of the measure theory of probability and a strict axiomatic system for the first time in his book The Basis of Probability Theory 1933. His axiomatic method became the basis of modern probability theory, making it a rigorous branch of mathematics, which played a positive role in the rapid development of probability theory.