Probability and Conditional Expectation

Rolf Steyer , Werner Nagel 지음

Wiley 2017년 02월 22일 출간

작품소개

책소개

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.

목차

Part I Measure-Theoretical Foundations of Probability Theory 1 Measure 3 1.1 Introductory Examples 3 1.2 σ-Algebra and Measurable Space 4 1.2.1 σ-Algebra Generated by a Set System 9 1.2.2 σ-Algebra of Borel Sets on Rn 12 1.2.3 σ-Algebra on a Cartesian Product 13 1.2.4 ∩-Stable Set Systems That Generate a σ-Algebra 15 1.3 Measure and Measure Space 16 1.3.1 σ-Additivity and Related Properties 17 1.3.2 Other Properties 18 1.4 Specific Measures 20 1.4.1 Dirac Measure and Counting Measure 21 1.4.2 Lebesgue Measure 22 1.4.3 Other Examples of a Measure 23 1.4.4 Finite and σ-Finite Measures 23 1.4.5 Product Measure 24 1.5 Continuity of a Measure 25 1.6 Specifying a Measure via a Generating System 27 1.7 σ-Algebra That is Trivial With Respect to a Measure 28 1.8 Proofs 28 1.9 Exercises 31 2 Measurable Mapping 41 2.1 Image and Inverse Image 41 2.2 Introductory Examples 42 2.2.1 Example 1: Rectangles 42 2.2.2 Example 2: Flipping two Coins 44 2.3 Measurable Mapping 46 2.3.1 Measurable Mapping 46 2.3.2 σ-Algebra Generated by a Mapping 51 2.3.3 Final σ-Algebra 54 2.3.4 Multivariate Mapping 54 2.3.5 Projection Mapping 56 2.3.6 Measurability With Respect to a Mapping 56 2.4 Theorems on Measurable Mappings 58 2.4.1 Measurability of a Composition 59 2.4.2 Theorems on Measurable Functions 61 2.5 Equivalence of Two Mappings With Respect to a Measure 64 2.6 Image Measure 67 2.7 Proofs 70 2.8 Exercises 75 3 Integral 83 3.1 Definition 83 3.1.1 Integral of a Nonnegative Step Function 83 3.1.2 Integral of a Nonnegative Measurable Function 88 3.1.3 Integral of a Measurable Function 93 3.2 Properties 96 3.2.1 Integral of μ-Equivalent Funct

작가정보

저자(글) Rolf Steyer

Rolf Steyer, Institute of Psychology, University of Jena, Germany Werner Nagel, Institute of Mathematics, University of Jena, Germany

저자(글) Werner Nagel