In this post, we will see the book Problems In Probability Theory, Mathematical Statistics And Theory Of Random Functions by A. A. Sveshnikov.

# About the book

Students at all levels of study in the theory of probability and in the theory of statistics will find in this book a broad and deep cross-section of problems (and their solutions) ranging from the simplest combinatorial probability problems in finite sample spaces through information theory, limit theorems and the use of moments.

The introductions to the sections in each chapter establish the basic formulas and notation and give a general sketch of that part of the theory that is to be covered by the problems to follow. Preceding each group of problems, there are typical examples and their solutions carried out in great detail. Each of these is keyed to the problems themselves so that a student seeking guidance in the solution of a problem can, by checking through the examples, discover the appropriate technique required for the solution.

The book was translated from Russian by Scripta Technica and edited by Bernard Gelbaum and was published in 1968.

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You can get the book here.

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# Contents

## I. RANDOM EVENT 1

1. Relations among random events l

2. A direct method for evaluating probabilities 4

3. Geometric probabilities 6

4. Conditional probability. The multiplication theorem for probabilites 12

5. The addition theorem for probabilities 16

6. The total probability formula 22

7. Computation of the probabilities of hypotheses after a trial (Bayes formula) 26

8. Evaluation of probabilities of occurrence of an event in repeated independent trials 30

9. The multinomial distribution. Recursion formulas. Generating functions 36

## II. RANDOM: VARIABLES 43

10. The probability distribution series, the distribution polygon and the distribution function of a discrete random variable 43

11. The distribution function and the probability density function of a continuous random variable 48

12. Numerical characteristics of discrete random variables 54

13. Numerical characteristics of continuous random variables 62

14. Poissons law 67

15. The normal distribution law 70

16. Characteristic functions 74

17. The computation of the total probability and the probability density in terms of conditional probability 80

## III. SYSTEMS OF RANDOM VARIABLES 84

18. Distribution laws and numerical characteristics of systems of random variables 84

19. The normal distribution law in the plane and in space. The multidimensional normal distribution 91

20. Distribution laws of subsystems of continuous random variables and conditional distribution laws 99

## IV. NUMERICAL CHARACTERISTICS AND DISTRIBUTION LAWS OF FUNCTIONS OF RANDOM VARIABLES 107

21. Numerical characteristics of functions of random variables .107

22. The distribution laws of functions of random variables. 115

23. The characteristic functions of systems and functions of random Variables 124

24. Convolution of distribution laws 128

25. The linearization of functions of random variables 136

26. The convolution of two-dimensional and three-dimensional normal distribution laws by use of the notion of deviation 145

## V. ENTROPY AND INFORMATION 157

27. The entropy of random events and variables 157

28. The quantity of information 163

## VI. THE LIMIT THEOREMS 171

29. The law of large numbers 171

30. The de Moivre-Laplace and Lyapunov theorems 176

## VII. THE CORRELATION THEORY OF RANDOM FUNCTIONS 181

31. General properties of correlation functions and distribution laws-of random functions 181

32. Linear operations with random functions. 185

33. Problems on Passages 192

34. Spectral decomposition of stationary random functions. 198

35. Computation of probability characteristics of random functions at the output of dynamical systems 205

36: Optimal-Dynamical systems 216

37: The method of envelopes 226

## VIII. MARKOV PROCESSES 231

38. Markov Chains 231

39. The Markov processes with a discrete number of states 246

40. Continuous Markov processes 256

## IX. METHODS OF DATA PROCESSING 275

41. Determination of the moments of random variables from experimental data 275

42. Confidence levels and confidence intervals 286

43; “Tests of goodness-of-fit 300

44. Data processing by the method of least squares 325

45. Statistical methods of quality control 346

46. Determination of probability characteristics of random functions from experimental data 368

ANSWERS: AND SOLUTIONS 375

SOURCES OF TABLES REFERRED TO IN THE TEXT 471

BIBLIOGRAPHY 475