An Introduction to probability theory and its applications/ William Feller

Includes index

Introduction: The Nature of probability theory -- The Sample space -- Elements of combinatorial analysis -- Fluctuations in coin tossing and random walks -- Combination of events -- Conditional probability stochastic independence -- The Binomial and the poisson distributions -- The Normal approximation to the binomial distribution -- Unlimited sequences of Bernouli trials -- Random variables; expectation -- Laws of large numbers -- Integral valued variables, generating functions -- Compound distributions, branching processes -- Recurrent events, renewal theory -- Random walk and ruin problems -- Markov chains -- Algebraic treatment of finite Markov chains -- The simplest time-dependent stochastic processes

A first introduction to the basic notations of probability is contained in Chapter I, V, VI, IX ; beginners should cover these with as few digressions as possible.
Chapter II is designed to develop the student's technique and probabilistic institution; some experience in its is desirable, but it is not necessary to cover the chapter systematically; it may prove more profitable to return to the elementary illustrations as occasion arises at later stages.
From Chapter IX an introductory courses may proceed directly to Chapter XI, considering generating as an example of more general transformations.
Chapter XI should be followed by some applications in chapter XIII (recurrent events) or XII (Chain reaction, infinitely divisible distributions).
Without generating functions it is possible to turn in one of the following directions: limit theorems and fluctuation theory (Chapter VIII, X, III); stochastic process (Chapter XVII); random walks (Chapter III) and the main part of XIV).
The markov chain chapter XV depend conceptually on recurrent events, but they may be studied independently if the reader is willing to accept without proof the basic ergodic theorem.
Chapter III stands by itself. Its contents are appealing in their own right, but the chapter is also highly illustrative for new insights and new methods in probability theory.