Sveshnikov A.A. (ed.) Probability Theory, Mathematical Statistics and Theory of Random Functions

Translated by Scripta Technica Inc. — Edited by B.R. Gelbaum. — Philadelphia; London; Toronto: W.B. Saunders Company, 1968. — 486 p.Students at all levels of study in the theory of probability and 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.Random events.
Relations among random events.
A direct method for evaluating probabilities.
Geometric probabilities.
Conditional probability. The multiplication theorem for probabilities.
The addition theorem for probabilities.
The total probability formula.
Computation of the probabilities of hypotheses after a trial (Bayes’ formula).
Evaluation of probabilities of occurrence of an event in repeated independent trials.
The multinomial distribution. Recursion formulas. Generating functions.
Random variables.
The probability distribution series, the distribution polygon, and the distribution function of a discrete random variable.
The distribution function and the probability density function of a continuous random variable.
Numerical characteristics of discrete random variables.
Numerical characteristics of continuous random variables Poisson’s law.
The normal distribution law.
Characteristic functions.
The computation of the total probability and the probability density in terms of conditional probability.
Systems of random variables.
Distribution laws and numerical characteristics of systems of random variables.
The normal distribution law in the plane and space. The multidimensional normal distribution.
Distribution laws of subsystems of continuous random variables and conditional distribution laws.
Numerical characteristics and distribution laws of functions of random variables.
Numerical characteristics of functions of random variables.
The distribution laws of functions of random variables.
The characteristic functions of systems and functions of random variables.
Convolution of distribution laws.
The linearization of functions of random variables.
The convolution of two-dimensional and three-dimensional normal distribution laws by use of the notion of deviation vectors.
Entropy and information.
The entropy of random events and variables.
The quantity of information.
The limit theorems.
The law of large numbers.
The de Moivre-Laplace and Lyapunov theorems.
The correlation theory of random functions
General properties of correlation functions and distribution laws of random functions.
Linear operations with random functions.
Problems with passages.
Spectral decomposition of stationary random functions.
Computation of probability characteristics of random functions at the output of dynamical systems.
Optimal dynamical systems.
The method of envelopes.
Markov processes.
Markov chains.
The Markov processes with a discrete number of states.
Continuous Markov processes.
Methods of data processing.
Determination of the moments of random variables from experimental data.
Confidence levels and confidence intervals.
Tests of goodness-of-fit.
Data processing by the method of least squares.
Statistical methods of quality control.
Determination of probability characteristics of random functions from experimental data.
Answers and solutions.
Sources of tables referred to in the text.