“Ygrek in Russian means the unknown,

to be calculated"

E. S. Wentzel

Every person's life reflects their time. And if it's a long life uncommon creative personality, then this reflection becomes more vivid and generally significant.

The life of Elena Sergeevna, 95 years long, embraced the whole existence Soviet power and ended at the beginning of the twenty-first century.

She was born in Reven (now Tallinn) in a family of teachers.

Father, Sergei Fedorovich Dolgintsev, taught mathematics, mother - literature. The family grew up two sons - Ilya and Nikolai - and daughter Elena. Sergei Fedorovich believed that higher mathematics was simpler than elementary and studied it with his daughter when she was 7-8 years old. As the writer herself recalls: “I went into mathematics because of my father. A mathematician by education, he dreamed of seeing at least one of his children continue his work. Of us, three children, I was perhaps the most adapted to this ... ”As a result, mathematics acquired Elena Sergeevna. In 1923, at the age of sixteen, she entered the Leningrad (then still Petrograd) University. The mathematical course, it should be noted, was made by men: out of 280 people, there were only five girls.

After graduating from the Faculty of Physics and Mathematics, in 1935 she moved to Moscow. She worked at the Air Force Academy. N. E. Zhukovsky (1935-68), at the Moscow Institute of Transport Engineers (1968-86); was engaged applied mathematics. In mathematics, she chose the most poetic section – the theory of probability. various kinds weapons, shooting at flying objects, air combat tactics, methods of organizing air defense systems. And her book The Theory of Probability remains to this day the most important textbook for engineers and students.

The list of scientific works of E. S. Wentzel includes about seventy open and sixty closed works. Military engineer, sailor, inventor N. V. Laptsevich wrote about her: “Her textbooks on probability theory and operations research ... belong to those ... very rare masterpieces, working through which you experience ... the joy of recognition and a sense of gratitude to the author ... ".

About how in the Academy. Zhukovsky professor and doctor technical sciences E. S. Wentzel gives lectures, there were rumors in all Moscow mathematical universities. The textbook on probability theory, written by her, was taken with pleasure not only in scientific, but also in public libraries: those who wished to win in« Sportloto » , calculate the existence of life on other planets, meet your destiny.

Engineer Victor Gastello, now newspaper correspondent« Shield and sword » , son legendary pilot recalled:

“She had a peculiar manner of presenting educational material. We called it diving. She constantly kept the audience in suspense. For example, explaining one of the sections of the theory of probability, she said this: “Imagine that a hundred monkeys are sitting in the audience (and there were about a hundred of us, the listeners), they all randomly knock on the keyboard. How likely are they to write a big Soviet Encyclopedia? (http://www.aif.ru/archive/1636620).

If Elena Sergeevna became an idol of listeners among mathematicians, then among writers the attention of young people was drawn to a certain I. Grekova, the author of the novel "The Department", the stories "The Widow's Steamboat", "The Ladies' Master" ... Few people knew that E. S. Wentzel and I. Grekova are one person.

In wonderful memories philologist Alexandra Alexandrovna Raskina, wife of the son of Elena Sergeevna, the world-famous mathematician Alexander Dmitrievich Wentzel, recorded the history of the emergence of a pseudonym, over which many readers racked their brains: “ When Tvardovsky was still going to print "Behind the Gateway", the question arose of a pseudonym. E.S. from the very beginning, she decided to strictly distinguish between these two incarnations of hers - a writer and a scientist (moreover, a teacher of a military academy). We sat at home, in the dining room and the whole family puzzled over this problem. Went on behalf of Elena. Yelenina? Yelenskaya? Tanya Wentzel remembered the Trojan Elena and says: Elena Grekova? And then E.S. suddenly exclaimed: “Igrekova!” And it immediately became clear that this was the way it should be.” A.A. Raskin "My Mother-in-Law" )

Elena Sergeevna herself recalls this as follows: “ In our family, there was a traditional interest in literature, we all wrote something. I started writing very early, publishing late. ...outwardly, I was a born mathematician. And internally, I was more drawn to literature. That's how my future life between mathematics and literature.

The release of the “public” works is to some extent connected with the name of Frida Abramovna Vigdorova (A.A. Raskina’s mother), having met and befriended whom, Elena Sergeevna brought her to read her story “Masters of Life”. A.A. Raskina: “ The story made us all very great impression, and not only a topic at all, although remember: it was the end of the sixtieth - two more years before "One Day in Ivan Denisovich"! But it was well written: for example, since then I remember that they serve “cold bluish cocoa” on the train. And it went. The next story was "The Yellow Flower". In the '66 story book, it's called "Under the Lantern" - like the rest of the book. And we all liked it too. But for some reason - maybe, by inertia, the first story was one hundred percent unprintable! - there was no talk that it would be necessary to print it. The "Yellow Flower" had an interesting fate. E.S. Few gave her stories to read. And my mother, if she liked something, gave it to all her friends to read - and there were many of them. And now the year 66 comes (mother was no longer there ...), collects E.S. storybook and can't find The Yellow Flower. Chernoviki E.S. I never kept it at all, and here I didn’t find a typewritten copy. And what did it turn out? Among those to whom my mother gave the story to read was the writer Raisa Orlova. Her daughter Masha, then still a schoolgirl, liked this story so much that she took it and copied it all by hand into a notebook. That's what a computer-free, but what's there - pre-Gutenberg, era we lived! Masha kept the notebook, the story was included in the book - in general, manuscripts do not burn!

She made her debut with the story "For the entrance" (1969), which, by the way, she wrote for her friend F.A. Vigdorova. A.A.Raskina: “But what is the story of the story “Behind the checkpoint.” His E.S. I wrote specifically for my mother, as they say, for internal use, so that my mother would get to know her, Elena Sergeevna, the environment, with her favorite scientists, “technicians”. … E.S. wrote it in the wake of a dispute about "physicists and lyricists" between the writer Ehrenburg and the engineer Poletaev, which captured almost the entire country. E.S., although she had already become a “lyricist”, was with the “physicists” in her soul and wanted to show them from the best side: the way she loved them with all her heart. And so, when the story was printed, she sent it to Ehrenburg with an inscription in Latin: “Audiatur et altera pars” – “Let the other side be heard.”

Criticism called her the "ruler of thoughts" of the educated part of the population in 1960-1970. Each new work of the writer was eagerly awaited, queues lined up for magazines in libraries. Few of the readers could know that the fate of both the writer herself and her works was very difficult, but very typical of the times of thaw and stagnation. She began writing early and publishing late. By the time of the publication of the story “Beyond the Gateway” (written in 1960), “the writer’s desk” already contained the stories “Masters of Life” (a story from a native Leningrader who fell into the “Kirov set”, passed through the camp, lost his family), “ Yellow flower”(known as“ Under the Lantern ”),“ The First Raid ”, the novel“ Fresh Tradition ”(the first name is“ Peano Curve ”) about the fate of Jews in Stalin's time. “The Masters of Life” became known to the reader 28 years later (in 1988), and the novel “Fresh Tradition” lay on the table for an “epic” period of “thirty years and three years (published in 1995 in the USA). The later story "Without Smiles" was able to see the light only 16 years later (in 1986). It took the writer more than 10 years to publish the story "The Widow's Steamboat" (written in the early 1970s, published in 1981).

As before, the most favorite works for the reader remain "The Chair", "The Widow's Steamboat", "The Hostess of the Inn".

The secret of the charm of I. Grekova's books is that they are always about people and the circumstances of their lives. They may or may not be successful, but they suffer, fight, believe or doubt. And always the main line is moral or search, or the choice of heroes.

Here is the assessment of I. Grekova given by S. Itskovich: “ a poet in mathematics and a mathematician in poetry, or rather, in prose, but after all, even her prose is poetic. She, according to Pushkin, seems to have managed to believe harmony with algebra: every word in her stories, novels, novels is verified and put in place with mathematical precision, like an X and a Y in a formula, which is why her prose sounds like a perfectly tuned musical instrument. ".

Treating language and plot with mathematical precision, she presented compressed texts and concise images that were infinitely close and understandable to readers. And even mathematical terms as metaphors revealed or supplemented the presented image. For example,

“Several people fussed with the luggage. Tall woman in trousers, compass spread her long legs, carefully moved the boxes with the instruments. I. Grekova shows us that there is no need to use a large number of metaphors, if you can get by with just one exact word.

“... it was not the difference that struck here, but identity<…>and as if she was here: unchanging, identical herself, the disturbing beauty of those two - the lady and the boy. In other words, identical means equal, similar. So here, when a military man shows a photo to the narrator, she instantly recognizes him and the woman next to him - this is his mother, with whom he looks similar.

I. Grekova herself was the embodiment of an ideal writer: she combined an excellent command of the material with excellent philological erudition. She was admired by many professional writers. FROM deep sadness accepted the news of the death of I. Grekov, a prose writer, head of the prose department of the magazine “ New world» Ruslan Kireev. Once he, an aspiring author, offered his story about a hairdresser to the magazine, not knowing that I. Grekova's "Ladies' Master" had already been accepted for publication. When, 15 years later, Kireev's text nevertheless came out, the already famous I. Grekova herself called him. According to Ruslan Kireev, he was always amazed by the erudition of the writer, who read Proust and Shakespeare in the original, quoting whole pages from Gogol by heart: “He was a man culture XIX century."

She passed away on April 15, 2002. Today her books, translated into different languages of the world are disappearing from the shelves just as quickly as before. After all, they are true and vital. When in the next Soviet journal her manuscript was asked to be corrected, she simply “took her offspring under her arm and left, even with a sense of relief - thank God, she wouldn’t have to cut, shred the living. Of course, if I lived on literary fees, I would be more accommodating, ”as Elena Sergeevna recalled. And in her declining years, she thanked fate, which saved her from immersing herself only in literature. After all, according to her, “there, as in any humanities of that time, it was necessary to “lie” in one form or another. But for us, mathematicians, it was easy to “live not by lies.”

"Harmonic combination of literature and exact sciences, impeccable professionalism and the same impeccable flair for falsehood in words and problem solving - this is the trademark of this person "- words of appreciation from" Nezavisimaya newspaper", 2002, April 19.

And here is what D. Bykov says and advises you, readers, today: “You see, Grekova, like [Vera] Panova, is such a miracle of style, very economical, very neutral, seasoned, calm, but at the same time capacious, and he already has a mind. You know, reading Grekova is like standing under a cold shower in the heat. The Ladies' Master is wonderful prose, so capacious and precise. And it is about the most important thing - about everyday everyday humiliation. There are a lot…”

March 16 was the birthday of Frida Abramovna Vigdorova (1915 - 1965), a talented writer, a brave journalist and kind, sympathetic person. Korney Ivanovich Chukovsky said about her: “The most best woman". Inscribing to her daughter Vigdorova, Sasha, her book “The Run of Time”, published in 1966, Anna Akhmatova called Frida Vigdorova “the highest example of kindness, nobility, humanity for all of us.” Read about it on our blog« The road to life Frida Vigdorova .

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1 E. S. Wentzel PROBABILITY THEORY Recommended by the Ministry of Education and Science Russian Federation as a textbook for students of higher technical educational institutions Eleventh edition, stereotypical 2010

2 UDC BBK V29 Reviewers: G. G. Olkhovsky, CEO All-Russian Thermal Engineering Institute, corresponding member. RAS, Dr. tech. Sciences, Prof. A. M. Petrova, Director of the Moscow polytechnic college, cand. economy Sciences, T. Yu. Simonova, Deputy. Director of the Moscow Polytechnic College Ventzel E.S. B29 Probability theory: textbook / E. S. Wentzel. 11th ed., ster. M. : KNORUS, p. ISBN The book is one of the best known textbooks on probability theory and is intended for those familiar with higher mathematics and interested in technical applications of probability theory. It is also of interest to those who apply the theory of probability in their practical activities. The book is given great attention various applications of probability theory (theory of probabilistic processes, information theory, queuing and etc.). For university students. Venttsel Elena Sergeevna PROBABILITY THEORY Sanitary and epidemiological conclusion D of the city. Ed. Signed for publication Format 60 90/16. Headset "NewtonC". Offset printing. Conv. oven l. 41.5. Uch. ed. l. 21.6. Circulation 3000 copies. Order. LLC "KnoRus Publishing House", Moscow, st. Bolshaya Pereyaslavskaya, 46, p. Goncharova, 14. UDC BBK Venttsel E. S. (heirs), 2010 CJSC "MCFER", 2010 ISBN LLC "Publishing House KnoRus", 2010

3 Contents Preface Chapter 1. Introduction 1.1. Subject of Probability Brief historical information Chapter 2. Basic concepts of probability theory 2.1. Event. Probability of an event Direct calculation of probabilities Frequency, or Statistical probability of an event Random value Practically impossible and practically certain events. Principle practical certainty Chapter 3. Basic theorems of probability theory 3.1. Purpose of the main theorems. Sum and product of events Probability addition theorem Probability multiplication theorem Formula full probability Hypothesis theorem (Bayes formula) Chapter 4. Repetition of experiments 4.1. Particular theorem on the repetition of experiments General theorem about repetition of experiments Chapter 5. Random variables and their laws of distribution 5.1. Distribution range. Distribution polygon Distribution function Probability of a random variable falling into a given area Distribution density Numerical characteristics random variables. Their role and purpose

4 4 Contents 5.6. Position characteristics (mathematical expectation, mode, median) Moments. Dispersion. Average standard deviation Law of uniform density Poisson's law Chapter 6. Normal distribution law 6.1. Normal law and its parameters Moments normal distribution The probability of hitting a random variable subordinated to normal law, to a given area. normal function distribution Probable (median) deviation Chapter 7. Determining the laws of distribution of random variables based on experimental data 7.1. Basic tasks of mathematical statistics Simple statistical population. Statistical distribution function Statistical series. Histogram Numerical characteristics statistical distribution alignment statistical series Goodness-of-fit criteria Chapter 8. Systems of random variables 8.1. The concept of a system of random variables Distribution function of a system of two random variables Distribution density of a system of two random variables Laws of distribution of individual variables included in the system. Conditional distribution laws Dependent and independent random variables Numerical characteristics of a system of two random variables. correlation moment. Correlation coefficient System of an arbitrary number of random variables Numerical characteristics of a system of several random variables

5 Chapter 9. Normal distribution law for a system of random variables Table of contents Normal law on the plane Scattering ellipses. Reduction of the normal law to the canonical form Probability of falling into a rectangle with sides parallel to the main dispersion axes Probability of falling into a scattering ellipse Probability of falling into an area of ​​arbitrary shape Normal law in the space of three dimensions. General entry normal law for a system of an arbitrary number of random variables Chapter 10. Numerical characteristics of functions of random variables Mathematical expectation of a function. The variance of the function numerical characteristics ax Applications of Theorems on Numerical Characteristics Chapter 11. Linearization of Functions Method of Linearization of Functions of Random Arguments Linearization of a Function of One random argument Linearization of a function of several random arguments Refinement of the results obtained by the linearization method Chapter 12. Laws of distribution of functions of random arguments distribution of the sum of two random variables. Composition of distribution laws Composition of normal laws Linear functions from normally distributed arguments Composition of normal laws on the plane

6 6 Table of contents Chapter 13. Limit theorems of probability theory Law big numbers and the central limit theorem Chebyshev's inequality Law of large numbers (Chebyshev's theorem) Generalized Chebyshev's theorem. Markov's theorem Consequences of the law of large numbers: Bernoulli's and Poisson's theorems Mass random phenomena and the central limit theorem Characteristic functions Central limit theorem for identically distributed terms Formulas expressing the central limit theorem and meeting with her practical application Chapter 14. Processing of experiments Features of processing a limited number of experiments. Estimates for unknown parameters of the distribution law Estimates for mathematical expectation and variance Confidence interval. Confidence probability Exact Construction Methods confidence intervals for parameters of a random variable distributed according to the normal law Probability estimation by frequency Estimates for the numerical characteristics of a system of random variables Shooting processing Smoothing experimental dependencies according to the method least squares Chapter 15 random function The concept of a random function as an extension of the concept of a system of random variables. Distribution law of a random function Characteristics of random functions Determining the characteristics of a random function from experience

7 Table of contents Methods for determining the characteristics of transformed random functions from the characteristics of the original random functions Linear and non-linear operators. Dynamic system operator Linear transformations random functions Addition of random functions Complex random functions Chapter 16. Canonical expansions of random functions The idea of ​​the method of canonical expansions. Representation of a random function as a sum of elementary random functions Canonical expansion of a random function Linear transformations of random functions given canonical expansions Chapter 17 Dispersion Spectrum Spectral expansion of a stationary random function over an infinite time interval. Spectral density stationary random function Spectral expansion of a random function in complex form Transformation of a stationary random function into a stationary one linear system Applications of the theory of stationary random processes to solving problems related to analysis and synthesis dynamic systems Ergodic property of stationary random functions Determining the characteristics of an ergodic stationary random function from one implementation Chapter 18. Basic concepts of information theory Subject and tasks of information theory Entropy as a measure of the degree of state uncertainty physical system

8 8 Contents Entropy complex system. Entropy addition theorem Conditional entropy. Combining Dependent Systems Entropy and Information Private information about a system contained in an event message. Private information about an event contained in a message about another event Entropy and information for systems with a continuous set of states Problems of message coding. Shannon Fano code Transmission of information with distortions. Bandwidth noisy channel Chapter 19. Elements of queuing theory Subject of queuing theory Random process with a countable set of states The flow of events. The simplest flow and its properties Unsteady Poisson flow Flow with limited aftereffect (Palm flow) Service time Markov stochastic process Queuing system with failures. Erlang Equations Steady Service Mode. Erlang formulas Waiting queuing system System mixed type with a limit on the length of the queue Appendix Index

9 Preface This book is written on the basis of lectures on the theory of probability given by the author over a number of years to students of the engineering academy them. N. E. Zhukovsky, as well as the author's textbook on the same subject. The textbook is designed mainly for an engineer with mathematical training in the volume of the usual course of higher technical educational institutions. When compiling the book, the author set the task of presenting the subject in the most simple and clear way, without tying himself within the framework of complete mathematical rigor. In this regard, certain provisions are given without proof (section on confidence limits and confidence probabilities; the theorem of A. N. Kolmogorov related to the criterion of agreement, and some others); some provisions are proved not quite rigorously (the theorem of multiplication of distribution laws; the rules for transforming the mathematical expectation and the correlation function when integrating and differentiating a random function, etc.). The applied mathematical apparatus, basically, does not go beyond the course higher mathematics, stated in higher technical educational institutions; where the author has to use less well-known concepts (for example, the concept linear operator, matrices, quadratic form etc.), these concepts are explained. The book is provided large quantity examples, in some cases of a calculated nature, in which the application of the methods presented is illustrated on a specific practical material and brought to a numerical result. Despite a somewhat specific selection of examples, the illustrative material contained in the book is understandable to engineers working in different areas technology, and anyone who uses the methods of probability theory in their work. The author is deeply grateful to Professor E. B. Dynkin and Professor V. S. Pugachev for a number of valuable suggestions. E. Wentzel

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Problems and exercises in probability theory. Ventzel E.S., Ovcharov L.A.

5th ed., rev. - M.: Academy, 2003.- 448 p..

This manual is a systematic collection of problems and exercises in probability theory. All problems are provided with answers, and most - with solutions. At the beginning of each chapter, a summary of the main theoretical positions and formulas needed to solve problems.

For students of higher technical educational institutions. It can be used by teachers, engineers and scientists interested in mastering probabilistic methods for solving practical problems.

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TABLE OF CONTENTS
Preface 3
Chapter 1. Basic concepts. Direct calculation of probabilities 4
Chapter 2. Theorems of addition and multiplication of probabilities 19
Chapter 3 Total Probability Formula and Bayes Formula 49
Chapter 4
Chapter 5. Random variables. Distribution laws. Numerical characteristics of random variables 85
Chapter 6. Systems of random variables (random vectors) 124
Chapter 7. Numerical characteristics of functions of random variables 152
Chapter 8. Laws of distribution of functions of random variables. Limit theorems of probability theory 207
Chapter 9 Random Functions 261
Chapter 10 Markov Stochastic Processes 317
Chapter 11 Queuing Theory 363
Applications 428
References 440

Name: Probability Theory. 1969.

The book is a textbook intended for people who are familiar with mathematics in the volume of a regular VTUZ course and are interested in technical applications of probability theory, in particular, the theory of shooting. The book is also of interest to engineers of other specialties who have to apply the theory of probability in their practical activities.
From other textbooks intended for the same category of readers, the book differs great attention to new branches of probability theory important for applications (for example, the theory of probabilistic processes, information theory, queuing theory, etc.).

Probability theory is mathematical science, studying patterns in random phenomena.
Let's agree what we mean by "random phenomenon".
At scientific research various physical in technical problems often have to deal with special type phenomena that are usually called random. A random phenomenon is such a phenomenon that, with repeated reproduction of the same experience, proceeds each time in a slightly different way.

TABLE OF CONTENTS
Preface to the second edition
Preface to the first edition 9
Chapter 1 Introduction 11
1.1. The subject of probability theory 11
1.2. Brief historical information 17
Chapter 2. Basic concepts of probability theory 23
2.1. Event. Probability of event 23
2.2. Direct calculation of probabilities 24
2.3. frequency, or statistical probability, events 28
2.4. Random value 32
2.5. Almost impossible and almost certain events. Principle of practical certainty 34
Chapter 3. Basic theorems of the theory of probability 37
3.1. Purpose of the main theorems. Sum and product of events 37
3.2. Probability addition theorem 40
3.3. Probability multiplication theorem 45
3.4. Total Probability Formula 54
3.5. Hypothesis theorem (Bayes formula) 56
Chapter 4
4.1. Particular theorem on the repetition of experiments 59
4.2. General theorem on repetition of experiments 61
Chapter 5. Random variables and their laws of distribution 67
5.1. Distribution range. Distribution polygon 67
5.2. Distribution function 72
5.3. Probability of hitting a random variable in a given area 78
5.4. Distribution density 80
5.5. Numerical characteristics of random variables. Their role and purpose 84
5.6. Position characteristics (mathematical expectation, mode, median) 85
5.7. Moments. Dispersion. Average standard deviation 92
5.8. Law of uniform density 103
5.9. Poisson's law. 106
Chapter 6
6.1. Normal law and its parameters 116
6.2. Normal distribution moments 120
6.3. The probability that a random variable obeying the normal law falls into a given area. Normal distribution function 122
6.4. Probable (median) deviation 127
Chapter 7. Determining the laws of distribution of random variables based on experimental data 131
7.1. Basic tasks of mathematical statistics 131
7.2. A simple statistic. Statistical distribution function 133
7.3. Statistical line. Histogram 133
7.4. Numerical characteristics of the statistical distribution 139
7.5. Flattening Statistical Series 143
7.6. Consent Criteria 149
Chapter 8. Systems of random variables 159
8.1. The concept of a system of random variables 159
8.2. Distribution function of a system of two random variables 163
8.3. Distribution Density of a System of Two Random Variables 163
8.4. Laws of distribution of individual quantities included in the system. Conditional laws of distribution 163
8.5. Dependent and independent random variables 171
8.6. Numerical characteristics of the system of two random values. correlation moment. Correlation coefficient 175
8.7. System of an arbitrary number of random variables 182
8.8. Numerical characteristics of a system of several random variables 184
Chapter 9. Normal distribution law for a system of random variables 188
9.1. Normal law on the plane 188
9.2. Scattering ellipses. Reduction of the normal law to the canonical form 193
9.3. Probability of hitting a rectangle with sides parallel to the main dispersion axes 196
9.4. Probability of hitting the dispersion ellipse 198
9.5. Probability of hitting a free-form area 202
9.6. Normal law in the space of three dimensions. General notation of the normal law for a system of an arbitrary number of random variables 205
Chapter 10. Numerical characteristics of functions of random variables 210
10.1. Mathematical expectation of a function. Function variance 210
10.2. Theorems on numerical characteristics 219
10.3. Applications of theorems on numerical characteristics 230
Chapter 11 Linearizing Functions 252
11.1. Linearization Method for Functions of Random Arguments 252
11.2. Linearization of a Function of One Random Argument 253
11.3. Linearizing a Function of Multiple Random Arguments 255
11.4. Refinement of the results obtained by the linearization method 259
Chapter 12. Laws of distribution of functions of random arguments 263
12.1. Distribution law of a monotonic function of one random argument 643
12.2. Distribution law of a linear function of an argument subject to the normal law 266
12.3. The distribution law of a nonmonotone function of one random argument 267
12.4. The law of distribution of a function of two random variables 269
12.5. The law of distribution of the sum of two random variables. Composition of laws of distribution 271
12.6. Composition of normal laws 275
12.7. Linear Functions of Normally Distributed Arguments 279
12.8. Composition of normal laws on the plane 280
Chapter 13
13.1. The Law of Large Numbers and the Central Limit Theorem 286
13.2. Chebyshev's inequality 28713.3. Law of large numbers (Chebyshev's theorem) 290
13.4. Generalized Chebyshev's theorem. Markov's theorem 292
13.5. Consequences of the Law of Large Numbers: Bernoulli and Poisson Theorems 295
13.6. Mass random phenomena and the central limit theorem 297
13.7. Characteristic functions 299
13.8. Central limit theorem for identically distributed terms 302
13.9. Formulas expressing the central limit theorem and encountered in its practical application 306
Chapter 14 Processing Experiences 312
14.1. Features of processing a limited number of experiments. Estimates for unknown parameters of the distribution law 312
14.2. Estimates for Expectation and Variance 314
14.3. Confidence interval. Confidence probability 317
14.4. Exact methods for constructing confidence intervals for the parameters of a random variable distributed according to the normal law 324
14.5. Frequency Probability Estimation 330
14.6. Estimates for the Numerical Characteristics of a System of Random Variables 339
14.7. Firing processing 347
14.8. Smoothing of experimental dependences by the method of least squares 351
Chapter 15. Basic concepts of the theory of random functions 370
15.1. The concept of a random function 370
15.2. The concept of a random function as an extension of the concept of a system of random variables. Distribution law of a random function 374
15.3. Characteristics of random functions 377
15.4. Determining the characteristics of a random function from experience 383
15.5. Methods for determining the characteristics of transformed random functions from the characteristics of the original random functions 385
15.6. Linear and non-linear operators. Dynamic System Operator 388
15.7. Linear transformations of random functions 393
15.8. Addition of random functions 39E
15.9. Complex random functions 402
Chapter 16. Canonical expansions of random functions 405
16.1. The idea of ​​the method of canonical expansions. Representation of a random function as a sum of elementary random functions 406
16.2. Canonical expansion of a random function 410
16.3. Linear transformations of random functions defined by canonical expansions 411
Chapter 17 Stationary Random Functions 419
17.1. The concept of a stationary random process 419
17.2. Spectral expansion of a stationary random function on a finite time interval. Dispersion spectrum 427
17.3. Spectral expansion of a stationary random function on an infinite time interval. Spectral density of a stationary random function 431
17.4. Spectral expansion of a random function in complex form 438
17.5. Transformation of a Stationary Random Function by a Stationary Linear System 447
17.6. Applications of the theory of stationary random processes to solving problems related to the analysis and synthesis of dynamic systems 454
17.7. Ergodic property of stationary random functions 457
17.8. Determining the characteristics of an ertodic stationary random function from one implementation 462
Chapter 18. Basic concepts of information theory 468
18.1. Subject and tasks, information theory 468
18.2. Entropy as a measure of the degree of uncertainty of the state of a physical system 469
18.3. Entropy of a complex system. Entropy addition theorem 475
15.1. Conditional entropy. Combining dependent systems 477
18.1. Entropy n information 481
18.2. Private information about the system contained in the event message. Private event information contained in another event message 489
18.7. Entropy and information for systems with a continuous set of states 493
18.8. Problems of message encoding. Shannon Code - Fano 502
18.9. Transmission of information with distortions. Noisy channel capacity 509
Chapter 19
19.1. Queuing Theory Subject 515
19.2. Random process with a countable set of states 517
19.3. The flow of events. The simplest flow and its properties 520
19.4. Unsteady Poisson flow 527
19. 5. Flow with limited aftereffect (Palma flow) 529
16. 6. Service time 534
19. 7. Markov Stochastic Process 537
19. 8. A queuing system with failures. Erlang Equations 540
19. 9. Steady mode of service. Erlang formulas 544
19.10. Waiting queuing system 548
19.11. Mixed system with limited queue length 557
Application. Tables 561
Literature 573
Index 574