Steyer R., Nagel W. Probability and Conditional Expectation: Fundamentals for the Empirical Sciences

John Wiley & Sons Ltd, UK, 2017. — 591 p. — (Wiley Series in Probability and Statistics) — ISBN: 9781119243489Probability 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.
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.Measure-Theoretical Foundations of Probability Theory
Measure
Measurable mapping
Integral
Probability, Random Variable, and its Distribution
Probability measure
Random variable, distribution, density, and distribution function
Expectation, variance, and other moments
Linear quasi-regression, covariance, and correlation
Some distributions
Conditional Expectation and Regression
Conditional expectation value and discrete conditional expectation
Conditional expectation
Residual, conditional variance, and conditional covariance
Linear regression
Linear logistic regression
Conditional expectation with respect to a conditional-probability measure
Effect functions of a discrete regressor
Conditional Independence and Conditional Distribution
Conditional independence
Conditional distribution