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

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

Published $\text {1968}$, Dover Publications

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


Translated from:

Sbornik Sadach po Teorii Veroyatnostey, Matematicheskoy Statistike i Teorii Sluchaynykh Funktsiy


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


Next


Further Editions


Errata

Target of Concentric Circles

$\text I$: Random Events: $1$. Relations among Random Events: Problem $3$

A target consists of $10$ concentric circles of radius $r_k (k = 1, 2, 3, \ldots, 10)$. An event $A_k$ means hitting the interior of a circle of radius $r_k (k = 1, 2, \ldots, 10)$. What do the following events mean?
$\ds B = \bigcup_{k \mathop = 1}^6 A_k, \qquad C = \prod_{k \mathop = 5}^{10} A_k$?


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