1.2. Brief historical information

Probability theory, like other mathematical sciences, has evolved from the needs of practice.

The beginning of a systematic study of problems related to mass random phenomena, and the emergence of the corresponding mathematical apparatus belong to the XVII century. At the beginning of the 17th century, the famous physicist Galileo tried to scientifically investigate the errors of physical measurements, treating them as random and evaluating their probabilities. The first attempts to create a general theory of insurance based on the analysis of patterns in such massive random events as morbidity, mortality, statistics of accidents, etc., belong to this time. The need to create a mathematical apparatus, specially adapted for the analysis of random phenomena, stemmed also from the needs of processing and generalizing extensive statistical material in all fields of science.

However, the theory of probability as a mathematical science was formed mainly not on the material of the above practical problems: these problems are too complex; in them, the laws governing random phenomena are not sufficiently visible and are obscured by many complicating factors. It was necessary to first study the patterns of random phenomena on a simpler material. Such material has historically been the so-called "gambling." From time immemorial, these games were created by a number of generations in such a way that in them the outcome of the experiment was independent of the observable conditions of experience, was purely random. The very word "passion" (fr. "Le hazard") means "case." Gambling schemes provide exceptional in their simplicity and transparency models of random phenomena, which allow them to observe and study the specific laws governing them in the most distinct form; and the possibility of unlimited repetition of the same experience provides an experimental verification of these laws under the conditions of actual mass character of the phenomena. Up to now, examples from the field of gambling and similar tasks for the “urn scheme” are widely used in studying the theory of probability as simplified models of random phenomena, illustrating the basic laws and rules of the theory of probability in the simplest and most visual form.

The emergence of probability theory in the modern sense of the word belongs to the middle of the XVII century and is associated with the research of Pascal (1623 - 1662), Fermat (1601 - 1665) and Huygens (1629 - 1695) in the field of gambling theory. In these works, such important concepts as probability and expectation were gradually formed; their basic properties and the methods of their calculation were established. Probabilistic methods have found direct practical application, first of all, in insurance problems. From the end of the 17th century, insurance began to be made on a scientific mathematical basis. Since then, probability theory has become increasingly widespread in various fields.

A major step forward in the development of the theory of probability is associated with the works of Jacob Bernoulli (1654 - 1705). He possesses the first proof of one of the most important provisions of the theory of probability, the so-called law of large numbers.

Even before Jacob Bernoulli, many noted as an empirical fact that feature of random phenomena, which can be called "the property of frequency stability with a large number of experiments." It was repeatedly noted that with a large number of experiments, the outcome of each of which is random, the relative frequency of occurrence of each given outcome tends to stabilize, approaching a certain specific number — the probability of this outcome. For example, if you repeatedly throw a coin, the relative frequency of the emblem approaches ½; With repeated throwing of a dice, the frequency of the appearance of a face with five points approaches 1/6, and so on. Jacob Bernoulli for the first time gave a theoretical justification for this empirical fact. Jacob Bernoulli's theorem — the simplest form of the law of large numbers — establishes a connection between the probability of an event and the frequency of its occurrence; with a sufficiently large number of experiments it is fashionable to expect with practical certainty an arbitrarily close coincidence of frequency with probability.

Another important stage in the development of the theory of probability is associated with the name of Moabra (1667 - 1754). This scientist was the first to introduce into consideration and, for the simplest case, substantiated a peculiar law, very often observed in random phenomena: the so-called normal law (otherwise, the Gauss law). Normal law, as we shall see, plays an extremely important role in random phenomena. The theorems that justify this law for certain conditions are, in probability theory, the general name of the “central limit theorem”.

An outstanding role in the development of probability theory belongs to the famous mathematician Laplace (1749 - 1827). He first gave a coherent and systematic exposition of the foundations of the theory of probability, gave proof of one of the forms of the central limit theorem (the Moavre – Laplace theorem) and developed a number of remarkable applications of probability theory to practical issues, in particular, to the analysis of errors in observations and measurements.

A significant step in the development of the theory of probability is associated with the name of Gauss (1777 - 1855), which gave an even more general justification to the normal law and developed a method for processing experimental data, known as the "least squares method." It should also be noted the work of Poisson (1781 - 1840), who proved a more general form than that of Jacob Bernoulli, the law of large numbers, and also first applied the theory of probability to shooting problems. The Poisson name is associated with one of the laws of distribution, which plays a large role in probability theory and its applications.

For the whole of the XVIII and the beginning of the XIX century, the rapid development of the theory of probability and its widespread fascination are characteristic. The theory of probability becomes a “fashionable” science. It is beginning to be applied not only where this application is legitimate, but also where it is not justified in any way. This period is characterized by numerous attempts to apply probability theory to the study of social phenomena, to the so-called "moral" or "moral" sciences. In a variety of works appeared on issues of justice, history, politics, even theology, which used the apparatus of the theory of probability. For all these pseudoscientific studies, an extremely simplified, mechanistic approach to the social phenomena considered in them is characteristic. The reasoning is based on some arbitrarily defined probabilities (for example, when considering issues of legal proceedings, the tendency of each person to truth or lie is estimated by some constant, equal probability for all people), and then the social problem is solved as a simple arithmetic problem. Naturally, all such attempts were doomed to failure and could not play a positive role in the development of science. On the contrary, their indirect result was that approximately in the 20s-30s of the 19th century in Western Europe, the widespread fascination with probability theory was replaced by disappointment and skepticism. Probability theory began to be viewed as a dubious, second-rate science, a kind of mathematical entertainment, hardly worthy of serious study.

It is remarkable that it was precisely at this time that the famous Petersburg Mathematical School was created in Russia, by whose works the theory of probability was put on a solid logical and mathematical basis and made a reliable, accurate and effective method of knowledge. Since the appearance of this school, the development of the theory of probability has already been closely associated with the work of Russians, and later of Soviet scientists.

Among students of the Petersburg Mathematical School should be called V. Ya. Bunyakovsky (1804 - 1889) - the author of the first course of probability theory in Russian, the creator of modern Russian terminology in probability theory, the author of original studies in statistics and demography.

The pupil of V. Ya. Bunyakovsky was the great Russian mathematician P. L. Chebyshev (1821 - 1894). Among the extensive and diverse mathematical works of P. L. Chebyshev, his works on probability theory occupy a prominent place. P. L. Chebyshev belongs to the further expansion and generalization of the law of large numbers. In addition, P. L. Chebyshev introduced into the theory of probability a very powerful and fruitful method of moments.

A pupil of P. L. Chebyshev was A. A. Markov (1856 - 1922), who also enriched probability theory with discoveries and methods of great importance. A. Markov significantly expanded the scope of the law of large numbers and the central limit theorem, extending them not only to independent but also to dependent experiments. The most important merit of A. A. Markov was that he laid the foundations of a completely new branch of the theory of probabilities — the theory of random, or “stochastic,” processes. The development of this theory is the main content of the latest, modern theory of probability.

A. M. Lyapunov (1857–1918) was also a pupil of P. L. Chebyshev, with whose name the first proof of the central limit theorem is associated under extremely general conditions. To prove his theorem, A. M. Lyapunov developed a special method of characteristic functions, widely used in modern probability theory.

A characteristic feature of the work of the St. Petersburg Mathematical School was the exceptional clarity of problem statement, the complete mathematical rigor of the methods used, and at the same time the close connection of the theory with the immediate requirements of practice. By the work of the scientists of the Petersburg Mathematical School, the theory of probability was derived from the backyard of science and placed as a full member in a series of exact mathematical sciences. The conditions for the application of its methods were strictly defined, and the methods themselves were brought to a high degree of perfection.

The modern development of the theory of probability is characterized by a general rise of interest in it and a sharp expansion of the range of its practical applications. Over the past decades, probability theory has become one of the most rapidly developing sciences, closely related to the needs of practice and technology. The Soviet school of probability theory, having inherited the traditions of the Petersburg Mathematical School, occupies a leading place in world science.

Here we will name only some of the largest Soviet scientists, whose works played a decisive role in the development of modern probability theory and its practical applications.

S. N. Bernstein developed the first complete axiomatic theory of probability, and also significantly expanded the scope of application of limit theorems.

A. Ya. Khinchin (1894 - 1959) is known for his research in the field of further generalization and strengthening of the law of large numbers, but mainly for his research in the field of so-called stationary random processes.

A number of the most important basic works in various fields of probability theory and mathematical statistics belong to A. N. Kolmogorov. He gave the most perfect axiomatic construction of the theory of probability, connecting it with one of the most important sections of modern mathematics - the metric theory of functions. Of particular importance are the works of A. N. Kolmogorov in the field of the theory of random functions (stochastic processes), which are currently the basis of all research in this field. The works of A. N. Kolmogorov relating to the evaluation of effectiveness formed the basis of a whole new scientific field in the theory of shooting, which later grew into a wider science on the effectiveness of military operations.

V. I. Romanovsky (1879 - 1954) and N. V. Smirnov are known for their work in the field of mathematical statistics, E. E. Slutsky (1880 - 1948) - in the theory of random processes, B. V. Gnedenko - in the field of mass theory servicing, E. B. Dynkin - in the field of Markov random processes, V. S. Pugachev - in the field of random processes as applied to automatic control problems.

The development of foreign probability theory is also going at an accelerated pace due to the imperative demands of practice. As with us, questions relating to random processes enjoy priority attention. Significant works in this area belong, for example, to N. Wiener, V. Feller, D. Oak. Important works on the theory of probability and mathematical statistics belong to R. Fisher, D. Neumann and G. Kramer.

In recent years, we have witnessed the birth of new and unique methods of applied probability theory, the appearance of which is associated with the specifics of the technical problems under study. It is, in particular, about such disciplines as "information theory" and "queuing theory". Emerged from the immediate needs of the practice, these sections of the theory of probability acquire a general theoretical value, and the range of their applications is constantly increasing.