# Book:A.A. Sveshnikov/Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions

## A.A. Sveshnikov: *Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions*

Published $1965$, **Dover Publications**

- ISBN 0-486-63717-4 (translated by Richard A. Silverman).

### Subject Matter

### Contents

- Foreword (Bernard R. Gelbaum)

- I. RANDOM EVENTS
- 1. Relations among random events
- 2. A direct method for evaluating probabilities
- 3. Geometric probabilities
- 4. Conditional probability. The multiplication theorem for probabilities
- 5. The addition theorem for probabilities
- 6. The total probability formula
- 7. Computation of the probabilities of hypotheses after a trial (Bayes' formula)
- 8. Evaluation of probabilities of occurrence of an event in repeated independent trials
- 9. The multinomial distribution. Recursion formulas. Generating functions

- II. RANDOM VARIABLES
- 10. The probability distribution series, the distribution polygon and the distribution function of a discrete random variable
- 11. The distribution function and the probability density function of a continuous random variable
- 12. Numerical characteristics of discrete random variables
- 13. Numerical characteristics of continuous random variables
- 14. Poisson's law
- 15. The normal distribution law
- 16. Characteristic functions
- 17. The computation of the total probability and the probability density in terms of conditional probability

- III. SYSTEMS OF RANDOM VARIABLES
- 18. Distribution laws and numerical characteristics of systems of random variables
- 19. The normal distribution law in the plane and in space. The multidimensional normal distribution
- 20. Distribution laws of subsystems of continuous random variables and conditional distribution laws

- IV. NUMERICAL CHARACTERISTICS AND DISTRIBUTION LAWS OF FUNCTIONS OF RANDOM VARIABLES
- 21. Numerical characteristics of functions of random variables
- 22. The distribution laws of functions of random variables
- 23. The characteristic functions of systems and functions of random variables
- 24. Convolution of distribution laws
- 25. The linearization of functions of random variables
- 26. The convolution of two-dimensional and three-dimensional normal distribution laws by use of the notion of deviation vectors

- V. ENTROPY AND INFORMATION
- 27. The entropy of random events and variables
- 28. The quantity of information

- VI. THE LIMIT THEOREMS
- 29. The law of large numbers
- 30. The de Moivre-Laplace and Lyapunov theorems

- VII. THE CORRELATION THEORY OF RANDOM FUNCTIONS
- 31. General properties of correlation functions and distribution laws of random functions
- 32. Linear operations with random functions
- 33. Problems on passages
- 34. Spectral decomposition of stationary random functions
- 35. Computation of probability characteristics of random functions at the output of dynamical systems
- 36. Optimal dynamical systems
- 37. The method of envelopes

- VIII. MARKOV PROCESSES
- 38. Markov Chains
- 39. The Markov processes with a discrete number of states
- 40. Continuous Markov processes

- IX. METHODS OF DATA PROCESSING
- 41. Determination of the moments of random variables from experimental data
- 42. Confidence levels and confidence intervals
- 43. Tests of goodness-of-fit
- 44. Data processing by the method of least squares
- 45. Statistical methods of quality control
- 46. Determination of probability characteristics of random functions from experimental data

- ANSWERS AND SOLUTIONS

- SOURCES OF TABLES REFERRED TO IN THE TEXT

- BIBLIOGRAPHY

- INDEX