Probability for Applications

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.