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Probability and Conditional Expectation : Fundamentals for the Empirical Sciences.

By: Contributor(s): Material type: TextTextSeries: Wiley Series in Probability and Statistics SeriesPublisher: Newark : John Wiley & Sons, Incorporated, 2017Copyright date: ©2017Edition: 1st edDescription: 1 online resource (601 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781119243502
Subject(s): Genre/Form: Additional physical formats: Print version:: Probability and Conditional ExpectationLOC classification:
  • QA273 .S749 2017
Online resources:
Contents:
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.
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 -- 14.5.1 Conditional expectation value with respect to PB -- 14.5.2 Uniqueness of factorizations -- 14.6 Uniqueness -- 14.6.1 A necessary and sufficient condition of uniqueness -- 14.6.2 Uniqueness with respect to P and other probability measures -- 14.6.3 Necessary and sufficient conditions of P-uniqueness -- 14.6.4 Properties related to P-uniqueness -- 14.7 Conditional mean independence with respect to PZ=z -- 14.8 Proofs -- 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 X and Z for regression coefficients -- 15.4 Adjusted effect functions -- 15.5 Logit effect functions -- 15.6 Implications of independence of X and Z for the logit regression coefficients -- 15.7 Proofs -- Part IV Conditional independence and conditional distribution -- 16 Conditional independence -- 16.1 Assumptions and definitions -- 16.1.1 Two events -- 16.1.2 Two sets of events -- 16.1.3 Two random variables -- 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 -- 17 Conditional distribution -- 17.1 Conditional distribution given a -algebra or a random variable -- 17.2 Conditional distribution given a value of a random variable -- 17.3 Existence and uniqueness -- 17.3.1 Existence -- 17.3.2 Uniqueness of the functions PY| (⋅, A′) -- 17.3.3 Common null set uniqueness of a conditional distribution -- 17.4 Conditional-probability measure given a value of a random variable.
17.5 Decomposing the joint distribution of random variables.
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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.

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 -- 14.5.1 Conditional expectation value with respect to PB -- 14.5.2 Uniqueness of factorizations -- 14.6 Uniqueness -- 14.6.1 A necessary and sufficient condition of uniqueness -- 14.6.2 Uniqueness with respect to P and other probability measures -- 14.6.3 Necessary and sufficient conditions of P-uniqueness -- 14.6.4 Properties related to P-uniqueness -- 14.7 Conditional mean independence with respect to PZ=z -- 14.8 Proofs -- 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 X and Z for regression coefficients -- 15.4 Adjusted effect functions -- 15.5 Logit effect functions -- 15.6 Implications of independence of X and Z for the logit regression coefficients -- 15.7 Proofs -- Part IV Conditional independence and conditional distribution -- 16 Conditional independence -- 16.1 Assumptions and definitions -- 16.1.1 Two events -- 16.1.2 Two sets of events -- 16.1.3 Two random variables -- 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 -- 17 Conditional distribution -- 17.1 Conditional distribution given a -algebra or a random variable -- 17.2 Conditional distribution given a value of a random variable -- 17.3 Existence and uniqueness -- 17.3.1 Existence -- 17.3.2 Uniqueness of the functions PY| (⋅, A′) -- 17.3.3 Common null set uniqueness of a conditional distribution -- 17.4 Conditional-probability measure given a value of a random variable.

17.5 Decomposing the joint distribution of random variables.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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