Introduction to Probability

This book was written for an introductory one-term course in probability. It is intended to provide the minimum background in probability that is necessary for students interested in applications to engineering and the sciences. Although it is aimed primarily at upperclassmen and beginning graduate students, the only prere quisite is the standard calculus course usually required of under graduates in engineering and science. Most beginning students will have some intuitive notions of the meaning of probability based on experiences involving, for example, games of chance. This book develops from these notions a set of precise and ordered concepts comprising the elementary theory of probability. An attempt has been made to state theorems carefully, but the level of the proofs varies greatly from formal arguments to appeals to intuition. The book is in no way intended as a substi tu te for a rigorous mathematical treatment of probability. How ever, some small amount of the language of formal mathematics is used, so that the student may become better prepared (at least psychologically) either for more formal courses or for study of the literature. Numerous examples are provided throughout the book. Many of these are of an elementary nature and are intended merely to illustrate textual material. A reasonable number of problems of varying difficulty are provided. Instructors who adopt the text for classroom use may obtain a Solutions Manual for all of the problems by writing to the author.

Contenu
1-Introduction and Preliminary Concepts.- 1.1-Random Phenomena.- 1.2-Elements of Set Theory.- 1.3-A Classical Concept of Probability.- 1.4-Elements of Combinatorial Analysis.- 1.5-The Axiomatic Foundation of Probability Theory.- 1.6-Finite Sample Spaces.- 1.7-Fields, ?-Fields, and Infinite Sample Spaces.- 1.8-Conditional Probability and Independence.- Problems.- 2-Random Variables.- 2.1-Definition.- 2.2-Discrete Random Variables.- 2.3-Continuous Random Variables.- 2.4-Random Vectors.- 2.5-Independence of Random Variables.- Problems.- 8-Distribution and Density Functions.- 3.1-Distribution Functions.- 3.2-Properties of Distribution Functions.- 3.3-The Decomposition of Distribution Functions.- 3.4-Discrete Distributions and Densities.- 3.5-Continuous Distributions and Densities.- 3.6-Mixed Distributions and Densities.- 3.7-Further Properties and Comments.- 3.8-Bivariate Distributions.- 3.9-Bivariate Density Functions.- 3.10-Multivariate Distributions.- 3.11-Independence.- 3.12-Conditional Distributions.- Problems.- 4-Expectations and Characteristic Functions.- 4.1-Expectation.- 4.2-Moments.- 4.3-The Bienayme Chebychev Theorem.- 4.4-The Moment Generating Function.- 4.5-The Chernoff Bound.- 4.6-The Characteristic Function.- 4.7-Covariances and Correlation Coefficients.- 4.8-Conditional Expectation.- 4.9-Least Mean-Squared Error Prediction.- Problems.- 5-The Binomial, Poisson, and Normal Distributions.- 5.1-The Binomial Distribution.- 5.2-The Poisson Distribution.- 5.3-The Normal or Gaussian Distribution.- 5.4-The Bivariate Normal Distribution.- 5.5-Rotations for Independence.- Problems.- 6-The Multivariate Normal Distribution.- 6.1-The Covariance Matrix.- 6.2-The Bivariate Normal Distribution in Matrix Form.- 6.3-The Multivariate Normal Distribution.- 6.4-MiscellaneousProperties of the Multivariate Normal Distribution.- 6.5-Linear Transformations on Normal Random Variables.- Problems.- 7-The Transformation of Random Variables.- 7.1-Discrete Random Variables.- 7.2-Continuous Random Variables; The Univariate Case.- 7.3-Continuous Random Variables; The Bivariate Case.- 7.4-Continuous Random Variables; A Special Case.- 7.5-Continuous Random Variables; The Multivariate Case.- Problems.- 8-Sequences of Random Variables.- 8.1-Convergence in the Deterministic Case.- 8.2-Convergence in Distribution.- 8.3-Convergence in Probability.- 8.4-Almost Sure Convergence.- 8.5-Convergence in r-th Mean.- 8.6-Relations Among Types of Convergence.- 8.7-Limit Theorems on Sums of Random Variables.- 8.8-The Weak Law of Large Numbers.- 8.9-The Strong Law of Large Numbers.- 8.10-The Central Limit Theorem.- Problems.- Appendices.- Appendix A The Riemann-Stieltjes Integral.- Appendix B The Dirac Delta Function.- Appendix C Interchange of Order in Differentiation and Integration.- Appendix D Some Elements of Matrix Theory.- Appendix E The Differentiation of a Definite Integral.- Appendix F A Monotone Non-Decreasing Function.- References.- Table I-The Unit Normal Distributions.