Collection of problems in probability theory

by L.D. Meshalkin

Overview

The Russian version of A collection of problems in probability theory contains a chapter devoted to statistics. That chapter has been omitted in this translation because, in the opinion of the editor, its content deviates somewhat from that which is suggested by the title: problems in pro­ bability theory. The original Russian version contains some errors; an attempt was made to correct all errors found, but perhaps a few stiII remain. An index has been added for the convenience of the reader who may be searching for a definition, a classical problem, or whatever. The index lists pages as well as problems where the indexed words appear. The book has been translated and edited with the hope of leaving as much "Russian flavor" in the text and problems as possible. Any pecu­ liarities present are most likely a result of this intention. August, 1972 Bryan A. Haworth viii Foreword to the Russian edition This Collection of problems in probability theory is primarily intended for university students in physics and mathematics departments. Its goal is to help the student of probability theory to master the theory more pro­ foundly and to acquaint him with the application of probability theory methods to the solution of practical problems. This collection is geared basically to the third edition of the GNEDENKO textbook Course in proba­ bility theory, Fizmatgiz, Moscow (1961), Probability theory, Chelsea (1965).

Product Details

ISBN-13:	9789401023603
Publisher:	Springer Netherlands
Publication date:	10/09/2011
Edition description:	Softcover reprint of the original 1st ed. 1973
Pages:	148
Product dimensions:	5.98(w) x 9.02(h) x 0.01(d)

Table of Contents

1 Fundamental concepts.- 1.1 Field of events.- 1.2 Interrelationships among cardinalities of sets.- 1.3 Definition of probability.- 1.4 Classical definition of probability. Combinatorics.- 1.5 Simplest problems on arrangements.- 1.6 Geometric probability.- 1.7 Metrization and ordering of sets.- 2 Application of the basic formulas.- 2.1 Conditional probability. Independence.- 2.2 Discrete distributions: binomial, multinomial, geometric, hypergeometric.- 2.3 Continuous distributions.- 2.4 Application of the formula for total probability.- 2.5 The probability of the sum of events.- 2.6 Setting up equations with the aid of the formula for total probability.- 3 Random variables and their properties.- 3.1 Calculation of mathematical expectations and dispersion.- 3.2 Distribution functions.- 3.3 Correlation coefficient.- 3.4 Chebyshev’s inequality.- 3.5 Distribution functions of random variables.- 3.6 Entropy and information.- 4 Basic limit theorems.- 4.1 The de Moivre-Laplace and Poisson theorems.- 4.2 Law of Large Numbers and convergence in probability.- 4.3 Central Limit Theorem.- 5 Characteristic and generating functions.- 5.1 Calculation of characteristic and generating functions.- 5.2 Connection with properties of a distribution.- 5.3 Use of the c.f. and g.f. to prove the limit theorems.- 5.4 Properties of c.f.’s and g.f.’s.- 5.5 Solution of problems with the aid of c.f.’s and g.f.’s.- 6 Application of measure theory.- 6.1 Measurability.- 6.2 Various concepts of convergence.- 6.3 Series of independent random variables.- 6.4 Strong law of large numbers and the iterated logarithm law.- 6.5 Conditional probabilities and conditional mathematical expectations.- 7 Infinitely divisible distributions. Normal law. Multidimensional distributions.- 7.1 Infinitelydivisible distributions.- 7.2 The normal distribution.- 7.3 Multidimensional distributions.- 8 Markov chains.- 8.1 Definition and examples. Transition probability matrix.- 8.2 Classification of states. Ergodicity.- 8.3 The distribution of random variables defined on a Markov chain.- Answers.- Suggested reading.