Approximately 1,000 problems &; with answers and solutions included at the back of the book &; illustrate such topics as random events, random variables, limit theorems, Markov processes, and much more.
I. RANDOM EVENTS 1. Relations among random events 2. A direct method for evaluating probabilities 3. Geometric probabilities 4. Conditonal 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 PROCESSNG 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
