Probability Theory

A Concise Course

Y. A. Rozanov

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486635446
• eBook ISBN: 9780486321141
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This book, a concise introduction to modern probability theory and certain of its ramifications, deals with a subject indispensable to natural scientists and mathematicians alike. Here the readers, with some knowledge of mathematics, will find an excellent treatment of the elements of probability together with numerous applications. Professor Y. A. Rozanov, an internationally known mathematician whose work in probability theory and stochastic processes has received wide acclaim, combines succinctness of style with a judicious selection of topics. His book is highly readable, fast-moving, and self-contained.
The author begins with basic concepts and moves on to combination of events, dependent events and random variables. He then covers Bernoulli trials and the De Moivre-Laplace theorem, which involve three important probability distributions (binomial, Poisson, and normal or Gaussian). The last three chapters are devoted to limit theorems, a detailed treatment of Markov chains, continuous Markov processes. Also included are appendixes on information theory, game theory, branching processes, and problems of optimal control. Each of the eight chapters and four appendixes has been equipped with numerous relevant problems (150 of them), many with hints and answers.
This volume is another in the popular series of fine translations from the Russian by Richard A. Silverman. Dr. Silverman, a former member of the Courant Institute of Mathematical Sciences of New York University and the Lincoln Laboratory of the Massachusetts Institute of Technology, is himself the author of numerous papers on applied probability theory. He has heavily revised the English edition and added new material. The clear exposition, the ample illustrations and problems, the cross-references, index, and bibliography make this book useful for self-study or the classroom.

1

BASIC CONCEPTS

1. Probability and Relative Frequency

Consider the simple experiment of tossing an unbiased coin. This experiment has two mutually exclusive outcomes, namely “heads” and “tails.” The various factors influencing the outcome of the experiment are too numerous to take into account, at least if the coin tossing is “fair.” Therefore the outcome of the experiment is said to be “random.” Everyone would certainly agree that the “probability of getting heads” and the “probability of getting tails” both equal

. Intuitively, this answer is based on the idea that the two outcomes are “equally likely” or “equiprobable,” because of the very nature of the experiment. But hardly anyone will bother at this point to clarify just what he means by “probability.”

Continuing in this vein and taking these ideas at face value, consider an experiment with a finite number of mutually exclusive outcomes which are equiprobable, i.e., “equally likely because of the nature of the experiment.” Let A denote some event associated with the possible outcomes of the experiment. Then the probability P(A) of the event A is defined as the fraction of the outcomes in which A occurs. More exactly,

where N is the total number of outcomes of the experiment and N(A) is the number of outcomes leading to the occurrence of the event A.

Example 1. In tossing a well-balanced coin, there are N = 2 mutually exclusive equiprobable outcomes (“heads” and “tails”). Let A be either of these two outcomes. Then N(A) = 1, and hence

Example 2. In throwing a single unbiased die, there are N = 6 mutually exclusive equiprobable outcomes, namely getting a number of spots equal to each of the numbers 1 through 6. Let A be the event consisting of getting an even number of spots. Then there are N(A) = 3 outcomes leading to the occurrence of A (which ones?), and hence

Example 3. In throwing a pair of dice, there are N = 36 mutually exclusive equiprobable events, each represented by an ordered pair (a, b), where a is the number of spots showing on the first die and b the number showing on the second die. Let A be the event that both dice show the same number of spots. Then A occurs whenever a = b, i.e., n(A) = 6. Therefore

Remark. Despite its seeming simplicity, formula (1.1) can lead to nontrivial calculations. In fact, before using (1.1) in a given problem, we must find all the equiprobable outcomes, and then identify all those leading to the occurrence of the event A in question.

The accumulated experience of innumerable observations reveals a remarkable regularity of behavior, allowing us to assign a precise meaning to the concept of probability not only in the case of experiments with equiprobable outcomes, but also in the most general case. Suppose the experiment under consideration can be repeated any number of times, so that, in principle at least, we can produce a whole series of “independent trials under identical conditions,”¹ in each of which, depending on chance, a partic...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• 1 Basic Concepts
• 2 Combination Of Events
• 3 Dependent Events
• 4 Random Variables
• 5 Three Important Probability Distributions
• 6 Some Limit Theorems
• 7 Markov Chains
• 8 Continuous Markov Processes
• Appendix 1 Information Theory
• Appendix 2 Game Theory
• Appendix 3 Branching Processes
• Appendix 4 Problems Of Optimal Control
• Bibliography
• Index