Pantsulaia G., Kvatadze Z., Giorgadze G. Elements of Probability Theory and Mathematical Statistics

Tbilisi: Georgian Technical University, 2013. — 188 p.The modern probability theory is an interesting and most important part of mathematics, which has great achievements and close connections both with classical parts of mathematics (geometry, mathematical analysis, functional analysis), and its various branches (theory of random processes, theory of ergodicity, theory of dynamical system, mathematical statistics and so on). The development of these branches of mathematics is mainly connected with the problems of statistical mechanics, statistical physics, statistical radio engineering and also with the problems of complicated systems which consider the random and the chaotic influence. At the origin of the probability theory were standing such famous mathematicians as I. Bernoulli, P. Laplace, S. Poisson, A. Cauchy, G. Cantor, F. Borel, A. Lebesgue and others. A very controversial problem connected with the relation between the probability theory and mathematics was entered in the list of unsolved mathematical problems raised by D. Hilbert in 1900. This problem has been solved by Russian mathematician A. Kolmogorov in 1933 who gave us a strict axiomatic basis of the probability theory. A. Kolmogorov's conception to the basis of the probability theory is applied in the present book. Giving a strong system of axioms (according to Kolmogorov) the general probability spaces and their coomposite components are described in the present book.
The main purpose of the present book is to help students to acquire such skills that are necessary to construct mathematical models (i.e., probability spaces) of various (social, economical, biological, mechanical, physical, etc.) processes and to calculate their numerical characteristics. In this sense the last chapters (in particular, chapters 14-15) are of interest, where some applications of various mathematical models (Markov chains, Brownian motion, etc.) are presented. The present book consists of twenty one chapters. More of chapters are equipped with exercises (i.e. tests), the solutions of which will help the student in deep comprehend and assimilation of experience of the presented elements of probability theory and mathematical statistics.Set-Theoretical Operations. Kolmogorov Axioms.
Properties of Probabilities.
Examples of Probability Spaces.
Total Probability and Bayes’ Formulas.
Applications of Caratheodory Method.
Random Variables.
Random variable distribution function.
Mathematical expectation and variance.
Correlation Coefficient.
Random Vector Distribution Function.
Chebishev’s inequalities.
Limit theorems.
The Method of Characteristic Functions and its applications.
Markov Chains.
The Process of Brownian Motion.
Mathematical Statistics.
Point, Well-Founded and Effective Estimations.
Point Estimators of Average and Variance.
Interval Estimation. Confidence intervals. Credible intervals. Interval Estimators of Parameters of Normally Distributed Random Variable.
Simple Hypothesis.
On consistent estimators of a useful signal in the linear one-dimensional stochastic model when an expectation of the transformed signal is not defined.True PDF