Probability

ISBN-13: 9781461269373 / Angielski / Miękka / 2012 / 283 str.

Alan F. Karr; Alan F. Karr

Opis

This book is a text at the introductory graduate level, for use in the one semester or two-quarter probability course for first-year graduate students that seems ubiquitous in departments of statistics, biostatistics, mathemat ical sciences, applied mathematics and mathematics. While it is accessi ble to advanced ("mathematically mature") undergraduates, it could also serve, with supplementation, for a course on measure-theoretic probability. Students who master this text should be able to read the "hard" books on probability with relative ease, and to proceed to further study in statistics or stochastic processes. This is a book to teach from. It is not encyclopredic, and may not be suitable for all reference purposes. Pascal once apologized to a correspondent for having written a long letter, saying that he hadn't the time to write a short one. I have tried to write a short book, which is quite deliberately incomplete, globally and locally. Many topics, including at least one of everyone's favorites, are omitted, among them, infinite divisibility, interchangeability, large devia tions, ergodic theory and the Markov property. These can be supplied at the discretion and taste of instructors and students, or to suit particular interests."

Spis treści

Prelude: Random Walks.- The Model.- Random variables.- Probability.- First calculations.- Issues and Approaches.- Issues.- Approaches.- Tools.- Functional of the Random Walk.- Times of returns to the origin.- Numbers of returns to the origin.- First passage times.- Maxima.- Time spent positive.- Limit Theorems.- Summary.- 1 Probability.- 1.1 Random Experiments and Sample Spaces.- 1.1.1 Random experiments.- 1.1.2 Sample spaces.- 1.2 Events and Classes of Sets.- 1.2.1 Events.- 1.2.2 Basic set operations.- 1.2.3 Indicator functions.- 1.2.4 Operations on sequences of sets.- 1.2.5 Classes of sets closed under set operations.- 1.2.6 Generated classes.- 1.2.7 The monotone class theorem.- 1.2.8 Events, bis.- 1.3 Probabilities and Probability Spaces.- 1.3.1 Probability.- 1.3.2 Elementary properties.- 1.3.3 More advanced properties.- 1.3.4 Almost sure and null events.- 1.3.5 Uniqueness.- 1.4 Probabilities on R.- 1.4.1 Distribution functions.- 1.4.2 Discrete probabilities.- 1.4.3 Absolutely continuous probabilities.- 1.4.4 Mixed distributions.- 1.5 Conditional Probability Given a Set.- 1.6 Complements.- 1.6.1 The extended real numbers.- 1.6.2 Measures.- 1.6.3 Lebesgue measure.- 1.6.4 Singular probabilities on R.- 1.6.5 Representation of probabilities on R.- 1.7 Exercises.- 2 Random Variables.- 2.1 Fundamentals.- 2.1.1 Random variables.- 2.1.2 Random vectors.- 2.1.3 Stochastic processes.- 2.1.4 Complex-valued random variables.- 2.1.5 The -algebra generated by a random variable.- 2.1.6 Simplified criteria.- 2.2 Combining Random Variables.- 2.2.1 Algebraic operations.- 2.2.2 Limiting operations.- 2.2.3 Transformations.- 2.2.4 Approximation of positive random variables.- 2.2.5 Monotone class theorems.- 2.3 Distributions and Distribution Functions.- 2.3.1 Random variables.- 2.3.2 Random vectors.- 2.4 Key Random Variables and Distributions.- 2.4.1 Discrete random variables.- 2.4.2 Absolutely continuous random variables.- 2.4.3 Random vectors.- 2.5 Transformation Theory.- 2.5.1 Random variables.- 2.5.2 Random vectors.- 2.6 Random Variables with Prescribed Distributions.- 2.6.1 Individual random variables.- 2.6.2 Random vectors.- 2.6.3 Sequences of random variables.- 2.7 Complements.- 2.7.1 Measurability with respect to sub-?-algebras.- 2.7.2 Borel measurable functions.- 2.8 Exercises.- 3 Independence.- 3.1 Independent Random Variables.- 3.1.1 Fundamentals.- 3.1.2 Criteria for independence.- 3.1.3 Examples.- 3.2 Functions of Independent Random Variables.- 3.2.1 Transformation properties.- 3.2.2 Sums of independent random variables.- 3.3 Constructing Independent Random Variables.- 3.3.1 Finite families.- 3.3.2 Sequences.- 3.4 Independent Events.- 3.5 Occupancy Models.- 3.5.1 Four occupancy models.- 3.5.2 Occupancy numbers.- 3.5.3 Asymptotics.- 3.6 Bernoulli and Poisson Processes.- 3.6.1 Bernoulli processes.- 3.6.2 Poisson processes.- 3.7 Complements.- 3.7.1 Independent ?-algebras.- 3.7.2 Products of probability spaces.- 3.8 Exercises.- 4 Expectation.- 4.1 Definition and Fundamental Properties.- 4.1.1 Simple random variables.- 4.1.2 Positive random variables.- 4.1.3 Integrable random variables.- 4.1.4 Complex-valued random variables.- 4.2 Integrals with respect to Distribution Functions.- 4.2.1 Generalities.- 4.2.2 Discrete distribution functions.- 4.2.3 Absolutely continuous distribution functions.- 4.2.4 Mixed distribution functions.- 4.3 Computation of Expectations.- 4.3.1 Positive random variables.- 4.3.2 Integrable random variables.- 4.3.3 Functions of random variables.- 4.3.4 Functions of random vectors.- 4.3.5 Functions of independent random variables.- 4.3.6 Sums of independent random variables.- 4.4 LP Spaces and Inequalities.- 4.4.1 LPspaces.- 4.4.2 Key inequalities.- 4.5 Moments.- 4.5.1 Moments of random variables.- 4.5.2 Variance and standard deviation.- 4.5.3 Covariance and correlation.- 4.5.4 Moments of random vectors.- 4.5.5 Multivariate normal distributions.- 4.6 Complements.- 4.6.1 Integration with respect to Lebesgue measure.- 4.6.2 Expectation for product probabilities.- 4.7 Exercises.- 5 Convergence of Sequences of Random Variables.- 5.1 Modes of Convergence.- 5.1.1 Convergence of random variables as functions.- 5.1.2 Convergence of distribution functions.- 5.1.3 Alternative criteria.- 5.2 Relationships Among the Modes.- 5.2.1 Implications always valid.- 5.2.2 Counterexamples.- 5.2.3 Implications of restricted validity.- 5.2.4 Implications involving subsequences.- 5.3 Convergence under Transformations.- 5.3.1 Algebraic operations.- 5.3.2 Continuous mappings.- 5.4 Convergence of Random Vectors.- 5.4.1 Convergence of random vectors as functions.- 5.4.2 Convergence in distribution.- 5.4.3 Continuous mappings.- 5.5 Limit Theorems for Bernoulli Summands.- 5.5.1 Laws of large numbers.- 5.5.2 Central limit theorems.- 5.5.3 The Poisson limit theorem.- 5.5.4 Approximation of continuous functions.- 5.6 Complements.- 5.6.1 LP Convergence of random variables.- 5.7 Exercises.- 6 Characteristic Functions.- 6.1 Definition and Basic Properties.- 6.1.1 Fundamentals.- 6.1.2 Elementary properties.- 6.2 Inversion and Uniqueness Theorems.- 6.2.1 The inversion theorem.- 6.2.2 The uniqueness theorem.- 6.2.3 Specialized inversion theorems.- 6.3 Moments and Taylor Expansions.- 6.3.1 Calculation of moments known to exist.- 6.3.2 Establishing existence of moments.- 6.3.3 Taylor expansions of characteristic functions.- 6.4 Continuity Theorems and Applications.- 6.4.1 Convergence in distribution.- 6.4.2 The Levy continuity theorem.- 6.4.3 Application to classical limit theorems.- 6.5 Other Transforms.- 6.5.1 Characteristic functions of random vectors.- 6.5.2 Laplace transforms.- 6.5.3 Moment generating functions.- 6.5.4 Generating functions.- 6.6 Complements.- 6.6.1 Helly’s theorem.- 6.7 Exercises.- 7 Classical Limit Theorems.- 7.1 Series of Independent Random Variables.- 7.1.1 Kolmogorov’s inequality.- 7.1.2 The three series theorem.- 7.2 The Strong Law of Large Numbers.- 7.3 The Central Limit Theorem.- 7.3.1 The Lyapunov condition.- 7.3.2 The Lindeberg condition.- 7.4 The Law of the Iterated Logarithm.- 7.4.1 Normally distributed summands.- 7.4.2 More general versions.- 7.5 Applications of the Limit Theorems.- 7.5.1 Monte Carlo integration.- 7.5.2 Maximum likelihood estimation.- 7.5.3 Empirical distribution functions.- 7.5.4 Random sums of independent random variables.- 7.5.5 Renewal processes.- 7.6 Complements.- 7.6.1 The Berry-Esseen theorem.- 7.7 Exercises.- 8 Prediction and Conditional Expectation.- 8.1 Prediction in L2.- 8.1.1 The inner product and norm.- 8.1.2 L2 as metric space.- 8.1.3 Orthogonality and orthonormality.- 8.1.4 The orthogonal decomposition theorem.- 8.1.5 Computation of MMSE predictors.- 8.1.6 Linear prediction.- 8.2 Conditional Expectation Given a Finite Set of Random Variables.- 8.2.1 Basics.- 8.2.2 Examples.- 8.2.3 Conditional probability.- 8.3 Conditional Expectation for X?L2.- 8.3.1 Conditional expectation as MMSE prediction.- 8.3.2 Properties of conditional expectation.- 8.4 Positive and Integrable Random Variables.- 8.5 Conditional Distributions.- 8.5.1 Generalities.- 8.5.2 Discrete random variables.- 8.5.3 Absolutely continuous random variables.- 8.6 Computational Techniques.- 8.6.1 General results.- 8.6.2 Special cases.- 8.7 Complements.- 8.7.1 Mixed conditional distributions.- 8.7.2 Conditional expectation given a ?-algebra.- 8.8 Exercises.- 9 Martingales.- 9.1 Fundamentals.- 9.1.1 Definitions.- 9.1.2 Examples.- 9.1.3 Compositions and transformations.- 9.2 Stopping Times.- 9.3 Optional Sampling Theorems.- 9.3.1 Optional sampling theorems for martingales.- 9.3.2 Applications of optional sampling theorems.- 9.4 Martingale Convergence Theorems.- 9.4.1 Upcrossings and almost sure convergence.- 9.4.2 Almost sure convergence of submartingales.- 9.4.3 Almost sure convergence of martingales.- 9.4.4 Uniformly integrable martingales.- 9.5 Applications of Convergence Theorems.- 9.5.1 The Radon-Nikodym theorem.- 9.5.2 Zero-one laws.- 9.5.3 Likelihood ratios.- 9.6 Complements.- 9.6.1 Conditioning on Yo,…,YT.- 9.6.2 Martingales with respect to filtrations.- 9.6.3 Reversed martingales.- 9.7 Exercises.- A Notation.- B Named Objects.