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

Subject Matter

 * Probability Theory
 * Statistics

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



Target of Concentric Circles
$\text I$: Random Events: $1$. Relations among Random Events: Problem $3$

Source work progress
* : $\text I$: Random Events: $1$. Relations among Random Events: Problem $4 \ \text {(a)}$