Probability for Applications

Objecti'ves. As the title suggests, this book provides an introduction to probability designed to prepare the reader for intelligent and resourceful applications in a variety of fields. Its goal is to provide a careful exposition of those concepts, interpretations, and analytical techniques needed for the study of such topics as statistics, introductory random processes, statis­ tical communications and control, operations research, or various topics in the behavioral and social sciences. Also, the treatment should provide a background for more advanced study of mathematical probability or math­ ematical statistics. The level of preparation assumed is indicated by the fact that the book grew out of a first course in probability, taken at the junior or senior level by students in a variety of fields-mathematical sciences, engineer­ ing, physics, statistics, operations research, computer science, economics, and various other areas of the social and behavioral sciences. Students are expected to have a working knowledge of single-variable calculus, including some acquaintance with power series. Generally, they are expected to have the experience and mathematical maturity to enable them to learn new concepts and to follow and to carry out sound mathematical arguments. While some experience with multiple integrals is helpful, the essential ideas can be introduced or reviewed rather quickly at points where needed.

I Basic Probability.- 1 Trials and Events.- 1.1 Trials, Outcomes, and Events.- 1.2 Combinations of Events and Special Events.- 1.3 Indicator Functions and Combinations of Events.- 1.4 Classes, Partitions, and Boolean Combinations.- 2 Probability Systems.- 2.1 Probability Measures.- 2.2 Some Elementary Properties.- 2.3 Interpretation and Determination of Probabilities.- 2.4 Minterm Maps and Boolean Combinations.- 2a The Sigma Algebra of Events.- 3 Conditional Probability.- 3.1 Conditioning and the Reassignment of Likelihoods.- 3.2 Properties of Conditional Probability.- 3.3 Repeated Conditioning.- 4 Independence of Events.- 4.1 Independence as a Lack of Conditioning.- 4.2 Independent Classes.- 5 Conditional Independence of Events.- 5.1 Operational Independence and a Common Condition.- 5.2 Equivalent Conditions and Definition.- 5.3 Some Problems in Probable Inference.- 5.4 Classification Problems.- 6 Composite Trials.- 6.1 Events and Component Events.- 6.2 Multiple Success-Failure Trials.- 6.3 Bernoulli Trials.- II Random Variables and Distributions.- 7 Random Variables and Probabilities.- 7.1 Random Variables as Functions—Mapping Concepts.- 7.2 Mass Transfer and Probability Distributions.- 7.3 Simple Random Variables.- 7a Borel Sets, Random Variables, and Borel Functions.- 8 Distribution and Density Functions.- 8.1 The Distribution Function.- 8.2 Some Discrete Distributions.- 8.3 Absolutely Continuous Random Variables and Density Functions.- 8.4 Some Absolutely Continuous Distributions.- 8.5 The Normal Distribution.- 8.6 Life Distributions in Reliability Theory.- 9 Random Vectors and Joint Distributions.- 9.1 The Joint Distribution Determined by a Random Vector.- 9.2 The Joint Distribution Function and Marginal Distributions.- 9.3 Joint Density Functions.- 10 Independence of Random Vectors.- 10.1 Independence of Random Vectors.- 10.2 Simple Random Variables.- 10.3 Joint Density Functions and Independence.- 11 Functions of Random Variables.- 11.1 A Fundamental Approach and some Examples.- 11.2 Functions of More Than One Random Variable.- 11.3 Functions of Independent Random Variables.- 11.4 The Quantile Function.- 11.5 Coordinate Transformations.- 11a Some Properties of the Quantile Function.- III Mathematical Expectation.- 12 Mathematical Expectation.- 12.1 The Concept.- 12.2 The Mean Value of a Random Variable.- 13 Expectation and Integrals.- 13.1 A Sketch of the Development.- 13.2 Integrals of Simple Functions.- 13.3 Integrals of Nonnegative Functions.- 13.4 Integrable Functions.- 13.5 Mathematical Expectation and the Lebesgue Integral.- 13.6 The Lebesgue-Stieltjes Integral and Transformation of Integrals.- 13.7 Some Further Properties of Integrals.- 13.8 The Radon-Nikodym Theorem and Fubini’s Theorem.- 13.9 Integrals of Complex Random Variables and the Vector Space ?2.- 13a Supplementary Theoretical Details.- 13a.1 Integrals of Simple Functions.- 13a.2 Integrals of Nonnegative Functions.- 13a.3 Integrable Functions.- 14 Properties of Expectation.- 14.1 Some Basic Forms of Mathematical Expectation.- 14.2 A Table of Properties.- 14.3 Independence and Expectation.- 14.4 Some Alternate Forms of Expectation.- 14.5 A Special Case of the Radon-Nikodym Theorem.- 15 Variance and Standard Deviation.- 15.1 Variance as a Measure of Spread.- 15.2 Some Properties.- 15.3 Variances for Some Common Distributions.- 15.4 Standardized Variables and the Chebyshev Inequality.- 16 Covariance, Correlation, and Linear Regression.- 16.1 Covariance and Correlation.- 16.2 Some Examples.- 16.3 Linear Regression.- 17 Convergence in Probability Theory.- 17.1 Sequences of Events.- 17.2 Almost Sure Convergence.- 17.3 Convergence in Probability.- 17.4 Convergence in the Mean.- 17.5 Convergence in Distribution.- 18 Transform Methods.- 18.1 Expectations and Integral Transforms.- 18.2 Transforms for Some Common Distributions.- 18.3 Generating Functions for Nonnegative, Integer-Valued Random Variables.- 18.4 Moment Generating Function and the Laplace Transform.- 18.5 Characteristic Functions.- 18.6 The Central Limit Theorem.- 18.7 Random Samples and Statistics.- IV Conditional Expectation.- 19 Conditional Expectation, Given a Random Vector.- 19.1 Conditioning by an Event.- 19.2 Conditioning by a Random Vector—Special Cases.- 19.3 Conditioning by a Random Vector—General Case.- 19.4 Properties of Conditional Expectation.- 19.5 Regression and Mean-Square Estimation.- 19.6 Interpretation in Terms of Hilbert Space ?2.- 19.7 Sums of Random Variables and Convolution.- 19a Some Theoretical Details.- 20 Random Selection and Counting Processes.- 20.1 Introductory Examples and a Formal Representation.- 20.2 Independent Selection from an lid Sequence.- 20.3 A Poisson Decomposition Result—Multinomial Trials.- 20.4 Extreme Values.- 20.5 Bernoulli Trials with Random Execution Times.- 20.6 Arrival Times and Counting Processes.- 20.7 Arrivals and Demand in an Independent Random Time Period.- 20.8 Decision Schemes and Markov Times.- 21 Poisson Processes.- 21.1 The Homogeneous Poisson Process.- 21.2 Arrivals of m kinds and compound Poisson processes.- 21.3 Superposition of Poisson Processes.- 21.4 Conditional Order Statistics.- 21.5 Nonhomogeneous Poisson Processes.- 21.6 Bibliographic Note.- 21a.- 21a.1 Independent Increments.- 21a.2 Axiom Systems for the Homogeneous Poisson Process.- 22 Conditional Independence, Given a Random Vector.- 22.1 The Concept and Some Basic Properties.- 22.2 The Bayesian Approach to Statistics.- 22.3 Elementary Decision Models.- 22a Proofs of Properties.- 23 Markov Sequences.- 23.1 The Markov Property for Sequences.- 23.2 Some Further Patterns and Examples.- 23.3 The Chapman-Kolmogorov Equation.- 23.4 The Transition Diagram and the Transition Matrix.- 23.5 Visits to a Given State in a Homogeneous Chain.- 23.6 Classification of States in Homogeneous Chains.- 23.7 Recurrent States and Limit Probabilities.- 23.8 Partitioning Finite Homogeneous Chains.- 23.9 Evolution of Finite, Ergodic Chains.- 23.10 The Strong Markov Property for Sequences.- 23a Some Theoretical Details.- A Some Mathematical Aids.- B Some Basic Counting Problems.