Probability and conditional expectation fundamentals for the empirical sciences 1st Edition by Rolf Steyer, Werner Nagel 1119243489 9781119243489

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ISBN-10 :  1119243489 

ISBN-13 : 9781119243489

Author:  Rolf Steyer, Werner Nagel 

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.

Probability and conditional expectation : fundamentals for the empirical sciences 1st table of contents:

Part I: Measure-theoretical foundations of probability theory
1: Measure
1.1 Introductory examples
1.2 σ-Algebra and measurable space
1.3 Measure and measure space
1.4 Specific measures
1.5 Continuity of a measure
1.6 Specifying a measure via a generating system
1.7 σ-Algebra that is trivial with respect to a measure
1.8 Proofs
Exercises
Solutions
2: Measurable mapping
2.1 Image and inverse image
2.2 Introductory examples
2.3 Measurable mapping
2.4 Theorems on measurable mappings
2.5 Equivalence of two mappings with respect to a measure
2.6 Image measure
2.7 Proofs
Exercises
Solutions
3: Integral
3.1 Definition
3.2 Properties
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
Exercises
Solutions
Part II: Probability, random variable, and its distribution
4: Probability measure
4.1 Probability measure and probability space
4.2 Conditional probability
4.3 Independence
4.4 Conditional independence given an event
4.5 Proofs
Exercises
Solutions
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
5.3 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
5.7 Uni- or multivariate real-valued random variable
5.8 Proofs
Exercises
Solutions
6: Expectation, variance, and other moments
6.1 Expectation
6.2 Moments, variance, and standard deviation
6.3 Proofs
Exercises
Solutions
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
7.5 Multiple linear quasi-regression
7.6 Proofs
Exercises
Solutions
8: Some distributions
8.1 Some distributions of discrete random variables
8.2 Some distributions of continuous random variables
8.3 Proofs
Exercises
Solutions
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
Exercises
Solutions
10: Conditional expectation
10.1 Assumptions and definitions
10.2 Existence and uniqueness
10.3 Rules of computation and other properties
10.4 Factorization, regression, and conditional expectation value
10.5 Characterizing a conditional expectation by the joint distribution
10.6 Conditional mean independence
10.7 Proofs
Exercises
Solutions
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 σ-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
Exercises
Solutions
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
Exercises
Solutions
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
Exercises
Solutions
14: Conditional expectation with respect to a conditional-probability measure
14.1 Introductory examples
14.2 Assumptions and definitions
14.3 Properties
14.4 Partial conditional expectation
14.5 Factorization
14.6 Uniqueness
14.7 Conditional mean independence with respect to
14.8 Proofs
Exercises
Solutions
15: Effect functions of a discrete regressor
15.1 Assumptions and definitions
15.2 Intercept function and effect functions
15.3 Implications of independence of and for regression coefficients
15.4 Adjusted effect functions
15.5 Logit effect functions
15.6 Implications of independence of and for the logit regression coefficients
15.7 Proofs
Exercises
Solutions
Part IV: Conditional independence and conditional distribution
16: Conditional independence
16.1 Assumptions and definitions
16.2 Properties
16.3 Conditional independence and conditional mean independence
16.4 Families of events
16.5 Families of set systems
16.6 Families of random variables
16.7 Proofs
Exercises
Solutions
17: Conditional distribution
17.1 Conditional distribution given a σ-algebra or a random variable
17.2 Conditional distribution given a value of a random variable
17.3 Existence and uniqueness
17.4 Conditional-probability measure given a value of a random variable
17.5 Decomposing the joint distribution of random variables
17.6 Conditional independence and conditional distributions
17.7 Expectations with respect to a conditional distribution
17.8 Conditional distribution function and probability density
17.9 Conditional distribution and Radon-Nikodym density
17.10 Proofs

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