{"product_id":"probability-and-conditional-expectation-9781119243526","title":"Probability and Conditional Expectation","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003ci\u003eProbability and Conditional Expectations \u003c\/i\u003ebridges 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eProbability and Conditional Expectations\u003c\/i\u003e\u003c\/p\u003e \u003cul\u003e \u003cli\u003ePresents a rigorous and detailed mathematical treatment of probability theory focusing on concepts that are fundamental to understand what we are estimating in applied statistics.\u003c\/li\u003e \u003cli\u003eExplores the basics of random variables along with extensive coverage of measurable functions and integration.\u003c\/li\u003e \u003cli\u003eExtensively treats conditional expectations also with respect to a conditional probability measure and the concept of conditional effect function\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003ePart I Measure-Theoretical Foundations of Probability Theory\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Measure 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introductory Examples 3\u003c\/p\u003e \u003cp\u003e1.2 σ-Algebra and Measurable Space 4\u003c\/p\u003e \u003cp\u003e1.2.1 σ-Algebra Generated by a Set System 9\u003c\/p\u003e \u003cp\u003e1.2.2 σ-Algebra of Borel Sets on Rn 12\u003c\/p\u003e \u003cp\u003e1.2.3 σ-Algebra on a Cartesian Product 13\u003c\/p\u003e \u003cp\u003e1.2.4 ∩-Stable Set Systems That Generate a σ-Algebra 15\u003c\/p\u003e \u003cp\u003e1.3 Measure and Measure Space 16\u003c\/p\u003e \u003cp\u003e1.3.1 σ-Additivity and Related Properties 17\u003c\/p\u003e \u003cp\u003e1.3.2 Other Properties 18\u003c\/p\u003e \u003cp\u003e1.4 Specific Measures 20\u003c\/p\u003e \u003cp\u003e1.4.1 Dirac Measure and Counting Measure 21\u003c\/p\u003e \u003cp\u003e1.4.2 Lebesgue Measure 22\u003c\/p\u003e \u003cp\u003e1.4.3 Other Examples of a Measure 23\u003c\/p\u003e \u003cp\u003e1.4.4 Finite and σ-Finite Measures 23\u003c\/p\u003e \u003cp\u003e1.4.5 Product Measure 24\u003c\/p\u003e \u003cp\u003e1.5 Continuity of a Measure 25\u003c\/p\u003e \u003cp\u003e1.6 Specifying a Measure via a Generating System 27\u003c\/p\u003e \u003cp\u003e1.7 σ-Algebra That is Trivial With Respect to a Measure 28\u003c\/p\u003e \u003cp\u003e1.8 Proofs 28\u003c\/p\u003e \u003cp\u003e1.9 Exercises 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Measurable Mapping 41\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Image and Inverse Image 41\u003c\/p\u003e \u003cp\u003e2.2 Introductory Examples 42\u003c\/p\u003e \u003cp\u003e2.2.1 Example 1: Rectangles 42\u003c\/p\u003e \u003cp\u003e2.2.2 Example 2: Flipping two Coins 44\u003c\/p\u003e \u003cp\u003e2.3 Measurable Mapping 46\u003c\/p\u003e \u003cp\u003e2.3.1 Measurable Mapping 46\u003c\/p\u003e \u003cp\u003e2.3.2 σ-Algebra Generated by a Mapping 51\u003c\/p\u003e \u003cp\u003e2.3.3 Final σ-Algebra 54\u003c\/p\u003e \u003cp\u003e2.3.4 Multivariate Mapping 54\u003c\/p\u003e \u003cp\u003e2.3.5 Projection Mapping 56\u003c\/p\u003e \u003cp\u003e2.3.6 Measurability With Respect to a Mapping 56\u003c\/p\u003e \u003cp\u003e2.4 Theorems on Measurable Mappings 58\u003c\/p\u003e \u003cp\u003e2.4.1 Measurability of a Composition 59\u003c\/p\u003e \u003cp\u003e2.4.2 Theorems on Measurable Functions 61\u003c\/p\u003e \u003cp\u003e2.5 Equivalence of Two Mappings With Respect to a Measure 64\u003c\/p\u003e \u003cp\u003e2.6 Image Measure 67\u003c\/p\u003e \u003cp\u003e2.7 Proofs 70\u003c\/p\u003e \u003cp\u003e2.8 Exercises 75\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Integral 83\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Definition 83\u003c\/p\u003e \u003cp\u003e3.1.1 Integral of a Nonnegative Step Function 83\u003c\/p\u003e \u003cp\u003e3.1.2 Integral of a Nonnegative Measurable Function 88\u003c\/p\u003e \u003cp\u003e3.1.3 Integral of a Measurable Function 93\u003c\/p\u003e \u003cp\u003e3.2 Properties 96\u003c\/p\u003e \u003cp\u003e3.2.1 Integral of μ-Equivalent Functions 98\u003c\/p\u003e \u003cp\u003e3.2.2 Integral With Respect to a Weighted Sum of Measures 100\u003c\/p\u003e \u003cp\u003e3.2.3 Integral With Respect to an Image Measure 102\u003c\/p\u003e \u003cp\u003e3.2.4 Convergence Theorems 103\u003c\/p\u003e \u003cp\u003e3.3 Lebesgue and Riemann Integral 104\u003c\/p\u003e \u003cp\u003e3.4 Density 106\u003c\/p\u003e \u003cp\u003e3.5 Absolute Continuity and the Radon-Nikodym Theorem 108\u003c\/p\u003e \u003cp\u003e3.6 Integral With Respect to a Product Measure 110\u003c\/p\u003e \u003cp\u003e3.7 Proofs 111\u003c\/p\u003e \u003cp\u003e3.8 Exercises 120\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II Probability, Random Variable and its Distribution\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Probability Measure 127\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Probability Measure and Probability Space 127\u003c\/p\u003e \u003cp\u003e4.1.1 Definition 127\u003c\/p\u003e \u003cp\u003e4.1.2 Formal and Substantive Meaning of Probabilistic Terms 128\u003c\/p\u003e \u003cp\u003e4.1.3 Properties of a Probability Measure 128\u003c\/p\u003e \u003cp\u003e4.1.4 Examples 130\u003c\/p\u003e \u003cp\u003e4.2 Conditional Probability 132\u003c\/p\u003e \u003cp\u003e4.2.1 Definition 132\u003c\/p\u003e \u003cp\u003e4.2.2 Filtration and Time Order Between Events and Sets of Events 133\u003c\/p\u003e \u003cp\u003e4.2.3 Multiplication Rule 135\u003c\/p\u003e \u003cp\u003e4.2.4 Examples 136\u003c\/p\u003e \u003cp\u003e4.2.5 Theorem of Total Probability 137\u003c\/p\u003e \u003cp\u003e4.2.6 Bayes’ Theorem 138\u003c\/p\u003e \u003cp\u003e4.2.7 Conditional-Probability Measure 139\u003c\/p\u003e \u003cp\u003e4.3 Independence 143\u003c\/p\u003e \u003cp\u003e4.3.1 Independence of Events 143\u003c\/p\u003e \u003cp\u003e4.3.2 Independence of Set Systems 144\u003c\/p\u003e \u003cp\u003e4.4 Conditional Independence Given an Event 145\u003c\/p\u003e \u003cp\u003e4.4.1 Conditional Independence of Events Given an Event 145\u003c\/p\u003e \u003cp\u003e4.4.2 Conditional Independence of Set Systems Given an Event 146\u003c\/p\u003e \u003cp\u003e4.5 Proofs 148\u003c\/p\u003e \u003cp\u003e4.6 Exercises 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Random Variable, Distribution, Density, and Distribution Function 155\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Random Variable and its Distribution 155\u003c\/p\u003e \u003cp\u003e5.2 Equivalence of Two Random Variables With Respect to a Probability Measure 161\u003c\/p\u003e \u003cp\u003e5.2.1 Identical and P-Equivalent Random Variables 161\u003c\/p\u003e \u003cp\u003e5.2.2 P-Equivalence, PB-Equivalence, and Absolute Continuity 164\u003c\/p\u003e \u003cp\u003e5.3 Multivariate Random Variable 167\u003c\/p\u003e \u003cp\u003e5.4 Independence of Random Variables 169\u003c\/p\u003e \u003cp\u003e5.5 Probability Function of a Discrete Random Variable 175\u003c\/p\u003e \u003cp\u003e5.6 Probability Density With Respect to a Measure 178\u003c\/p\u003e \u003cp\u003e5.6.1 General Concepts and Properties 178\u003c\/p\u003e \u003cp\u003e5.6.2 Density of a Discrete Random Variable 180\u003c\/p\u003e \u003cp\u003e5.6.3 Density of a Bivariate Random Variable 180\u003c\/p\u003e \u003cp\u003e5.7 Uni- or Multivariate Real-Valued Random Variable 182\u003c\/p\u003e \u003cp\u003e5.7.1 Distribution Function of a Univariate Real-Valued Random Variable 182\u003c\/p\u003e \u003cp\u003e5.7.2 Distribution Function of a Multivariate Real-Valued Random Variable 184\u003c\/p\u003e \u003cp\u003e5.7.3 Density of a Continuous Univariate Real-Valued Random Variable 185\u003c\/p\u003e \u003cp\u003e5.7.4 Density of a Continuous Multivariate Real-Valued Random Variable 187\u003c\/p\u003e \u003cp\u003e5.8 Proofs 188\u003c\/p\u003e \u003cp\u003e5.9 Exercises 196\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Expectation, Variance, and Other Moments 199\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Expectation 199\u003c\/p\u003e \u003cp\u003e6.1.1 Definition 199\u003c\/p\u003e \u003cp\u003e6.1.2 Expectation of a Discrete Random Variable 200\u003c\/p\u003e \u003cp\u003e6.1.3 Computing the Expectation Using a Density 202\u003c\/p\u003e \u003cp\u003e6.1.4 Transformation Theorem 203\u003c\/p\u003e \u003cp\u003e6.1.5 Rules of Computation 206\u003c\/p\u003e \u003cp\u003e6.2 Moments, Variance, and Standard Deviation 207\u003c\/p\u003e \u003cp\u003e6.3 Proofs 212\u003c\/p\u003e \u003cp\u003e6.4 Exercises 213\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Linear Quasi-Regression, Covariance, and Correlation 217\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Linear Quasi-Regression 217\u003c\/p\u003e \u003cp\u003e7.2 Covariance 220\u003c\/p\u003e \u003cp\u003e7.3 Correlation 224\u003c\/p\u003e \u003cp\u003e7.4 Expectation Vector and Covariance Matrix 227\u003c\/p\u003e \u003cp\u003e7.4.1 Random Vector and Random Matrix 227\u003c\/p\u003e \u003cp\u003e7.4.2 Expectation of a Random Vector and a Random Matrix 228\u003c\/p\u003e \u003cp\u003e7.4.3 Covariance Matrix of two Multivariate Random Variables 229\u003c\/p\u003e \u003cp\u003e7.5 Multiple Linear Quasi-Regression 231\u003c\/p\u003e \u003cp\u003e7.6 Proofs 233\u003c\/p\u003e \u003cp\u003e7.7 Exercises 237\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Some Distributions 245\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Some Distributions of Discrete Random Variables 245\u003c\/p\u003e \u003cp\u003e8.1.1 Discrete Uniform Distribution 245\u003c\/p\u003e \u003cp\u003e8.1.2 Bernoulli Distribution 246\u003c\/p\u003e \u003cp\u003e8.1.3 Binomial Distribution 247\u003c\/p\u003e \u003cp\u003e8.1.4 Poisson Distribution 250\u003c\/p\u003e \u003cp\u003e8.1.5 Geometric Distribution 252\u003c\/p\u003e \u003cp\u003e8.2 Some Distributions of Continuous Random Variables 254\u003c\/p\u003e \u003cp\u003e8.2.1 Continuous Uniform Distribution 254\u003c\/p\u003e \u003cp\u003e8.2.2 Normal Distribution 256\u003c\/p\u003e \u003cp\u003e8.2.3 Multivariate Normal Distribution 259\u003c\/p\u003e \u003cp\u003e8.2.4 Central χ2-Distribution 262\u003c\/p\u003e \u003cp\u003e8.2.5 Central t -Distribution 264\u003c\/p\u003e \u003cp\u003e8.2.6 Central F-Distribution 266\u003c\/p\u003e \u003cp\u003e8.3 Proofs 267\u003c\/p\u003e \u003cp\u003e8.4 Exercises 271\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart III Conditional Expectation and Regression\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Conditional Expectation Value and Discrete Conditional Expectation 277\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Conditional Expectation Value 277\u003c\/p\u003e \u003cp\u003e9.2 Transformation Theorem 280\u003c\/p\u003e \u003cp\u003e9.3 Other Properties 282\u003c\/p\u003e \u003cp\u003e9.4 Discrete Conditional Expectation 283\u003c\/p\u003e \u003cp\u003e9.5 Discrete Regression 285\u003c\/p\u003e \u003cp\u003e9.6 Examples 287\u003c\/p\u003e \u003cp\u003e9.7 Proofs 291\u003c\/p\u003e \u003cp\u003e9.8 Exercises 291\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Conditional Expectation 295\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Assumptions and Definitions 295\u003c\/p\u003e \u003cp\u003e10.2 Existence and Uniqueness 297\u003c\/p\u003e \u003cp\u003e10.2.1 Uniqueness With Respect to a Probability Measure 298\u003c\/p\u003e \u003cp\u003e10.2.2 A Necessary and Sufficient Condition of Uniqueness 299\u003c\/p\u003e \u003cp\u003e10.2.3 Examples 300\u003c\/p\u003e \u003cp\u003e10.3 Rules of Computation and Other Properties 301\u003c\/p\u003e \u003cp\u003e10.3.1 Rules of Computation 301\u003c\/p\u003e \u003cp\u003e10.3.2 Monotonicity 302\u003c\/p\u003e \u003cp\u003e10.3.3 Convergence Theorems 302\u003c\/p\u003e \u003cp\u003e10.4 Factorization, Regression, and Conditional Expectation Value 306\u003c\/p\u003e \u003cp\u003e10.4.1 Existence of a Factorization 306\u003c\/p\u003e \u003cp\u003e10.4.2 Conditional Expectation and Mean-Squared Error 307\u003c\/p\u003e \u003cp\u003e10.4.3 Uniqueness of a Factorization 308\u003c\/p\u003e \u003cp\u003e10.4.4 Conditional Expectation Value 309\u003c\/p\u003e \u003cp\u003e10.5 Characterizing a Conditional Expectation by the Joint Distribution 312\u003c\/p\u003e \u003cp\u003e10.6 Conditional Mean Independence 313\u003c\/p\u003e \u003cp\u003e10.7 Proofs 318\u003c\/p\u003e \u003cp\u003e10.8 Exercises 321\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Residual, Conditional Variance, and Conditional Covariance 329\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Residual With Respect to a Conditional Expectation 329\u003c\/p\u003e \u003cp\u003e11.2 Coefficient of Determination and Multiple Correlation 333\u003c\/p\u003e \u003cp\u003e11.3 Conditional Variance and Covariance Given a σ-Algebra 338\u003c\/p\u003e \u003cp\u003e11.4 Conditional Variance and Covariance Given a Value of a Random Variable 339\u003c\/p\u003e \u003cp\u003e11.5 Properties of Conditional Variances and Covariances 342\u003c\/p\u003e \u003cp\u003e11.6 Partial Correlation 345\u003c\/p\u003e \u003cp\u003e11.7 Proofs 347\u003c\/p\u003e \u003cp\u003e11.8 Exercises 348\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Linear Regression 357\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Basic Ideas 357\u003c\/p\u003e \u003cp\u003e12.2 Assumptions and Definitions 359\u003c\/p\u003e \u003cp\u003e12.3 Examples 361\u003c\/p\u003e \u003cp\u003e12.4 Linear Quasi-Regression 366\u003c\/p\u003e \u003cp\u003e12.5 Uniqueness and Identification of Regression Coefficients 367\u003c\/p\u003e \u003cp\u003e12.6 Linear Regression 369\u003c\/p\u003e \u003cp\u003e12.7 Parametrizations of a Discrete Conditional Expectation 370\u003c\/p\u003e \u003cp\u003e12.8 Invariance of Regression Coefficients 374\u003c\/p\u003e \u003cp\u003e12.9 Proofs 375\u003c\/p\u003e \u003cp\u003e12.10Exercises 377\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Linear Logistic Regression 381\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Logit Transformation of a Conditional Probability 381\u003c\/p\u003e \u003cp\u003e13.2 Linear Logistic Parametrization 383\u003c\/p\u003e \u003cp\u003e13.3 A Parametrization of a Discrete Conditional Probability 385\u003c\/p\u003e \u003cp\u003e13.4 Identification of Coefficients of a Linear Logistic Parametrization 387\u003c\/p\u003e \u003cp\u003e13.5 Linear Logistic Regression and Linear Logit Regression 388\u003c\/p\u003e \u003cp\u003e13.6 Proofs 394\u003c\/p\u003e \u003cp\u003e13.7 Exercises 396\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Conditional Expectation With Respect to a Conditional-Probability Measure 399\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Introductory Examples 399\u003c\/p\u003e \u003cp\u003e14.2 Assumptions and Definitions 404\u003c\/p\u003e \u003cp\u003e14.3 Properties 410\u003c\/p\u003e \u003cp\u003e14.4 Partial Conditional Expectation 412\u003c\/p\u003e \u003cp\u003e14.5 Factorization 413\u003c\/p\u003e \u003cp\u003e14.5.1 Conditional Expectation Value With Respect to PB 414\u003c\/p\u003e \u003cp\u003e14.5.2 Uniqueness of Factorizations 415\u003c\/p\u003e \u003cp\u003e14.6 Uniqueness 415\u003c\/p\u003e \u003cp\u003e14.6.1 A Necessary and Sufficient Condition of Uniqueness 415\u003c\/p\u003e \u003cp\u003e14.6.2 Uniqueness w.r.t. P and Other Probability Measures 417\u003c\/p\u003e \u003cp\u003e14.6.3 Necessary and Sufficient Conditions of P-Uniqueness 418\u003c\/p\u003e \u003cp\u003e14.6.4 Properties Related to P-Uniqueness 420\u003c\/p\u003e \u003cp\u003e14.7 Conditional Mean Independence With Respect to PZ=z 424\u003c\/p\u003e \u003cp\u003e14.8 Proofs 426\u003c\/p\u003e \u003cp\u003e14.9 Exercises 431\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Conditional Effect Functions of a Discrete Regressor 437\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Assumptions and Definitions 437\u003c\/p\u003e \u003cp\u003e15.2 Conditional Intercept Function and Effect Functions 438\u003c\/p\u003e \u003cp\u003e15.3 Implications of Independence of X and Z for Regression Coefficients 441\u003c\/p\u003e \u003cp\u003e15.4 Adjusted Conditional Effect Functions 443\u003c\/p\u003e \u003cp\u003e15.5 Conditional Logit Effect Functions 447\u003c\/p\u003e \u003cp\u003e15.6 Implications of Independence of X and Z for the Logit Regression Coefficients 450\u003c\/p\u003e \u003cp\u003e15.7 Proofs 452\u003c\/p\u003e \u003cp\u003e15.8 Exercises 454\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart IV Conditional Independence and Conditional Distribution\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 Conditional Independence 459\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Assumptions and Definitions 459\u003c\/p\u003e \u003cp\u003e16.1.1 Two Events 459\u003c\/p\u003e \u003cp\u003e16.1.2 Two Sets of Events 461\u003c\/p\u003e \u003cp\u003e16.1.3 Two Random Variables 462\u003c\/p\u003e \u003cp\u003e16.2 Properties 463\u003c\/p\u003e \u003cp\u003e16.3 Conditional Independence and Conditional Mean Independence 470\u003c\/p\u003e \u003cp\u003e16.4 Families of Events 473\u003c\/p\u003e \u003cp\u003e16.5 Families of Set Systems 473\u003c\/p\u003e \u003cp\u003e16.6 Families of Random Variables 475\u003c\/p\u003e \u003cp\u003e16.7 Proofs 478\u003c\/p\u003e \u003cp\u003e16.8 Exercises 486\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Conditional Distribution 491\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Conditional Distribution Given a σ-Algebra or a Random Variable 491\u003c\/p\u003e \u003cp\u003e17.2 Conditional Distribution Given a Value of a Random Variable 494\u003c\/p\u003e \u003cp\u003e17.3 Existence and Uniqueness 497\u003c\/p\u003e \u003cp\u003e17.3.1 Existence 497\u003c\/p\u003e \u003cp\u003e17.3.2 Uniqueness of the Functions PY |C ( ·, A′) 498\u003c\/p\u003e \u003cp\u003e17.3.3 Common Null Set (CNS) Uniqueness of a Conditional Distribution 499\u003c\/p\u003e \u003cp\u003e17.4 Conditional-Probability Measure Given a Value of a Random Variable 502\u003c\/p\u003e \u003cp\u003e17.5 Decomposing the Joint Distribution of Random Variables 504\u003c\/p\u003e \u003cp\u003e17.6 Conditional Independence and Conditional Distributions 506\u003c\/p\u003e \u003cp\u003e17.7 Expectations With Respect to a Conditional Distribution 511\u003c\/p\u003e \u003cp\u003e17.8 Conditional Distribution Function and Probability Density 513\u003c\/p\u003e \u003cp\u003e17.9 Conditional Distribution and Radon-Nikodym Density 516\u003c\/p\u003e \u003cp\u003e17.10Proofs 520\u003c\/p\u003e \u003cp\u003e17.11Exercises 536\u003c\/p\u003e \u003cp\u003eReferences 541\u003c\/p\u003e\n\u003c\/li\u003e\n\u003c\/ul\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49528850841943,"sku":"9781119243526","price":85.95,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781119243526.jpg?v=1731873272","url":"https:\/\/bookcurl.com\/products\/probability-and-conditional-expectation-9781119243526","provider":"Book Curl","version":"1.0","type":"link"}