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Probability and conditional expectation : fundamentals for the empirical sciences / Rolf Steyer, Werner Nagel
| データ種別 | 電子ブック |
|---|---|
| 出版者 | (Chichester, West Sussex : John Wiley & Sons, Inc) |
| 出版年 | 2017 |
| 大きさ | 1 online resource |
| 著者標目 | *Steyer, Rolf 1950- Nagel, Werner |
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| 一般注記 | Includes index Print version record Probability and Conditional Expectations bridges the gap between books on probability theory and statistics by providing the probabilistic concepts estimated and tested in analysis of variance, regression analysis, factor analysis, structural equation modeling, hierarchical linear models and analysis of qualitative data. The authors emphasize the theory of conditional expectations that is also fundamental to conditional independence and conditional distributions. Probability and Conditional Expectations -Presents a rigorous and detailed mathematical treatment of probability theory focusing on concepts that are fundamental to understand what we are estimating in applied statistics.-Explores the basics of random variables along with extensive coverage of measurable functions and integration.-Extensively treats conditional expectations also with respect to a conditional probability measure and the concept of conditional effect functions, which are crucial in the analysis of causal effects.-Is illustrated throughout with simple examples, numerous exercises and detailed solutions.-Provides website links to further resources including videos of courses delivered by the authors as well as R code exercises to help illustrate the theory presented throughout the book Includes bibliographical references and indexes Intro -- Probability and Conditional Expectation -- Contents -- Preface -- Why another book on probability? -- What is it about? -- For whom is it? -- Prerequisites -- Acknowledgements -- About the companion website -- Part I Measure-theoretical foundations of probability theory -- 1 Measure -- 1.1 Introductory examples -- 1.2 -Algebra and measurable space -- 1.2.1 -Algebra generated by a set system -- 1.2.2 -Algebra of Borel sets on -- 1.2.3 -Algebra on a Cartesian product -- 1.2.4)"Stable set systems that generate a -algebra -- 1.3 Measure and measure space -- 1.3.1 -Additivity and related properties -- 1.3.2 Other properties -- 1.4 Specific measures -- 1.4.1 Dirac measure and counting measure -- 1.4.2 Lebesgue measure -- 1.4.3 Other examples of a measure -- 1.4.4 Finite and -finite measures -- 1.4.5 Product measure -- 1.5 Continuity of a measure -- 1.6 Specifying a measure via a generating system -- 1.7 -Algebra that is trivial with respect to a measure -- 1.8 Proofs -- 2 Measurable mapping -- 2.1 Image and inverse image -- 2.2 Introductory examples -- 2.2.1 Example 1: Rectangles -- 2.2.2 Example 2: Flipping two coins -- 2.3 Measurable mapping -- 2.3.1 Measurable mapping -- 2.3.2 -Algebra generated by a mapping -- 2.3.3 Final -algebra -- 2.3.4 Multivariate mapping -- 2.3.5 Projection mapping -- 2.3.6 Measurability with respect to a mapping -- 2.4 Theorems on measurable mappings -- 2.4.1 Measurability of a composition -- 2.4.2 Theorems on measurable functions -- 2.5 Equivalence of two mappings with respect to a measure -- 2.6 Image measure -- 2.7 Proofs -- 3 Integral -- 3.1 Definition -- 3.1.1 Integral of a nonnegative step function -- 3.1.2 Integral of a nonnegative measurable function -- 3.1.3 Integral of a measurable function -- 3.2 Properties -- 3.2.1 Integral of -equivalent functions 3.2.2 Integral with respect to a weighted sum of measures -- 3.2.3 Integral with respect to an image measure -- 3.2.4 Convergence theorems -- 3.3 Lebesgue and Riemann integral -- 3.4 Density -- 3.5 Absolute continuity and the Radon-Nikodym theorem -- 3.6 Integral with respect to a product measure -- 3.7 Proofs -- Part II Probability, Random Variable, and Its Distribution -- 4 Probability measure -- 4.1 Probability measure and probability space -- 4.1.1 Definition -- 4.1.2 Formal and substantive meaning of probabilistic terms -- 4.1.3 Properties of a probability measure -- 4.1.4 Examples -- 4.2 Conditional probability -- 4.2.1 Definition -- 4.2.2 Filtration and time order between events and sets of events -- 4.2.3 Multiplication rule -- 4.2.4 Examples -- 4.2.5 Theorem of total probability -- 4.2.6 Bayes' theorem -- 4.2.7 Conditional-probability measure -- 4.3 Independence -- 4.3.1 Independence of events -- 4.3.2 Independence of set systems -- 4.4 Conditional independence given an event -- 4.4.1 Conditional independence of events given an event -- 4.4.2 Conditional independence of set systems given an event -- 4.5 Proofs -- 5 Random variable, distribution, density, and distribution function -- 5.1 Random variable and its distribution -- 5.2 Equivalence of two random variables with respect to a probability measure -- 5.2.1 Identical and P-equivalent random variables -- 5.2.2 P-equivalence, PB-equivalence, and absolute continuity -- 5.3 Multivariate random variable -- 5.4 Independence of random variables -- 5.5 Probability function of a discrete random variable -- 5.6 Probability density with respect to a measure -- 5.6.1 General concepts and properties -- 5.6.2 Density of a discrete random variable -- 5.6.3 Density of a bivariate random variable -- 5.7 Uni- or multivariate real-valued random variable 5.7.1 Distribution function of a univariate real-valued random variable -- 5.7.2 Distribution function of a multivariate real-valued random variable -- 5.7.3 Density of a continuous univariate real-valued random variable -- 5.7.4 Density of a continuous multivariate real-valued random variable -- 5.8 Proofs -- 6 Expectation, variance, and other moments -- 6.1 Expectation -- 6.1.1 Definition -- 6.1.2 Expectation of a discrete random variable -- 6.1.3 Computing the expectation using a density -- 6.1.4 Transformation theorem -- 6.1.5 Rules of computation -- 6.2 Moments, variance, and standard deviation -- 6.3 Proofs -- 7 Linear quasi-regression, covariance, and correlation -- 7.1 Linear quasi-regression -- 7.2 Covariance -- 7.3 Correlation -- 7.4 Expectation vector and covariance matrix -- 7.4.1 Random vector and random matrix -- 7.4.2 Expectation of a random vector and a random matrix -- 7.4.3 Covariance matrix of two multivariate random variables -- 7.5 Multiple linear quasi-regression -- 7.6 Proofs -- 8 Some distributions -- 8.1 Some distributions of discrete random variables -- 8.1.1 Discrete uniform distribution -- 8.1.2 Bernoulli distribution -- 8.1.3 Binomial distribution -- 8.1.4 Poisson distribution -- 8.1.5 Geometric distribution -- 8.2 Some distributions of continuous random variables -- 8.2.1 Continuous uniform distribution -- 8.2.2 Normal distribution -- 8.2.3 Multivariate normal distribution -- 8.2.4 Central 2-distribution -- 8.2.5 Central t-distribution -- 8.2.6 Central F-distribution -- 8.3 Proofs -- Part III Conditional expectation and regression -- 9 Conditional expectation value and discrete conditional expectation -- 9.1 Conditional expectation value -- 9.2 Transformation theorem -- 9.3 Other properties -- 9.4 Discrete conditional expectation -- 9.5 Discrete regression -- 9.6 Examples -- 9.7 Proofs -- 10 Conditional expectation 10.1 Assumptions and definitions -- 10.2 Existence and uniqueness -- 10.2.1 Uniqueness with respect to a probability measure -- 10.2.2 A necessary and sufficient condition of uniqueness -- 10.2.3 Examples -- 10.3 Rules of computation and other properties -- 10.3.1 Rules of computation -- 10.3.2 Monotonicity -- 10.3.3 Convergence theorems -- 10.4 Factorization, regression, and conditional expectation value -- 10.4.1 Existence of a factorization -- 10.4.2 Conditional expectation and mean squared error -- 10.4.3 Uniqueness of a factorization -- 10.4.4 Conditional expectation value -- 10.5 Characterizing a conditional expectation by the joint distribution -- 10.6 Conditional mean independence -- 10.7 Proofs -- 11 Residual, conditional variance, and conditional covariance -- 11.1 Residual with respect to a conditional expectation -- 11.2 Coefficient of determination and multiple correlation -- 11.3 Conditional variance and covariance given a -algebra -- 11.4 Conditional variance and covariance given a value of a random variable -- 11.5 Properties of conditional variances and covariances -- 11.6 Partial correlation -- 11.7 Proofs -- 12 Linear regression -- 12.1 Basic ideas -- 12.2 Assumptions and definitions -- 12.3 Examples -- 12.4 Linear quasi-regression -- 12.5 Uniqueness and identification of regression coefficients -- 12.6 Linear regression -- 12.7 Parameterizations of a discrete conditional expectation -- 12.8 Invariance of regression coefficients -- 12.9 Proofs -- 13 Linear logistic regression -- 13.1 Logit transformation of a conditional probability -- 13.2 Linear logistic parameterization -- 13.3 A parameterization of a discrete conditional probability -- 13.4 Identification of coefficients of a linear logistic parameterization -- 13.5 Linear logistic regression and linear logit regression -- 13.6 Proofs John Wiley and Sons Wiley Online Library: Complete oBooks HTTP:URL=https://onlinelibrary.wiley.com/doi/book/10.1002/9781119243496 |
|---|---|
| 件 名 | LCSH:Random variables LCSH:Measure theory LCSH:Measure algebras LCSH:Probabilities CSHF:Variables al�eatoires CSHF:Th�eorie de la mesure CSHF:Alg�ebres de mesures CSHF:Probabilit�es FREE:probability FREE:MATHEMATICS -- Applied 全ての件名で検索 FREE:MATHEMATICS -- Probability & Statistics -- General 全ての件名で検索 FREE:Measure algebras FREE:Measure theory FREE:Probabilities FREE:Random variables FREE:Wahrscheinlichkeitsrechnung FREE:Wahrscheinlichkeitstheorie FREE:Statistik |
| 分 類 | LCC:QA273 DC23:519.2 |
| 書誌ID | EB00004479 |
| ISBN | 9781119243502 |
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