Collection of problems in probability theory


Collection of problems in probability theory

L.D. Meshalkin ; translated from the Russian and ed. by Leo F. Boron and Bryan A. Haworth

Noordhoff International Pub., 1973


Sbornik zadach po teorii veroi︠a︡tnosteĭ

大学図書館所蔵 件 / 43



Translation of Sbornik zadach po teorii veroi︠a︡tnosteĭ

Bibliography: p. 145



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).


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 Infinitely divisible 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.

「Nielsen BookData」 より