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| Preface | |
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| Introduction | |
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| The Basic Model | |
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| Part I Introduction | |
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| Dealing with 'Real-World' Problems | |
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| The Probabilistic Experiment | |
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| Outcome | |
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| Events | |
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| The Connection to the Mathematical World | |
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| Elements and Sets | |
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| Classes of Sets | |
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| Elementary Set Operations | |
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| Additional Set Operations | |
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| Functions | |
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| The Size of a Set | |
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| Multiple and Infinite Set Operations | |
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| More About Additive Classes | |
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| Additive Set Functions | |
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| More about Probabilistic Experiments | |
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| The Probability Function | |
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| Probability Space | |
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| Simple Probability Arithmetic | |
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| Part I Summary | |
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| The Approach to Elementary Probability Problems | |
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| Part II Introduction | |
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| About Probability Problems | |
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| Equally Likely Possible Outcomes | |
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| Conditional Probability | |
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| Conditional Probability Distributions | |
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| Independent Events | |
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| Classes of Independent Events | |
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| Possible Outcomes Represented as Ordered k-Tuples | |
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| Product Experiments and Product Spaces | |
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| Product Probability Spaces | |
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| Dependence Between the Components in an Ordered k-Tuple | |
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| Multiple Observations Without Regard to Order | |
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| Unordered Sampling with Replacement | |
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| More Complicated Discrete Probability Problems | |
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| Uncertainty and Randomness | |
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| Fuzziness | |
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| Part II Summary | |
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| Introduction to Random Variables | |
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| Part III Introduction | |
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| Numerical-Valued Outcomes | |
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| The Binomial Distribution | |
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| The Real Numbers | |
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| General Definition of a Random Variable | |
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| The Cumulative Distribution Function | |
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| The Probability Density Function | |
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| The Gaussian Distribution | |
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| Two Discrete Random Variables | |
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| Two Arbitrary Random Variables | |
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| Two-Dimensional Distribution Functions | |
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| Two-Dimensional Density Functions | |
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| Two Statistically Independent Random Variables | |
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| Two Statistically Independent Random Variables-Absolutely Continuous Case | |
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| Part III Summary | |
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| Transformations and Multiple Random Variables | |
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| Part IV Introduction | |
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| Transformation of a Random Variable | |
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| Transformation of a discrete random variable | |
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| Transformation of an arbitrary random variable | |
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| Transformation of an absolutely continuous random variable | |
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| Transformation of a Two-Dimensional Random Variable | |
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| The Sum of Two Discrete Random Variables | |
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| The Sum of Two Arbitrary Random Variables | |
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| n-Dimensional Random Variables | |
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| Absolutely Continuous n-Dimensional R.V.'s | |
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| Coordinate Transformations | |
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| Rotations and the Bivariate Gaussian Distribution | |
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| Several Statistically Independent Random Variables | |
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| Singular Distributions in One Dimension | |
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| Conditional Induced Distribution, Given an Event | |
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| Resolving a Distribution into Components of Pure Type | |
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| Conditional Distribution Given the Value of a Random Variable | |
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| Random Occurrences in Time | |
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| Part IV Summary | |
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| Parameters for Describing Random Variables and Induced Distributions | |
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| Part V Introduction | |
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| Some Properties of a Random Variable | |
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| Higher Moments | |
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| Expectation of a Function of a Random Variable | |
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| Scale change and shift of origin | |
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| General formulation | |
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| Sum of random variables | |
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| Powers of a random variable | |
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| Product of random variables | |
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| The Variance of a Function of a Random Variable | |
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| Bounds on the Induced Distribution | |
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| Test Sampling | |
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| A Simple random sample | |
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| Unbiased estimators | |
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| Variance of the sample average | |
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| Estimating the population variance | |
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| Sampling with replacement | |
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| Conditional Expectation with Respect to an Event | |
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| Covariance and Correlation Coefficient | |
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| The Correlation Coefficient as Parameter in a Joint Distribution | |
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| More General Kinds of Dependence Between Random Variables | |
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| The Covariance Matrix | |
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| Random Variables as the Elements of a Vector Space | |
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| Estimation | |
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| The concept of estimating a random variable | |
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| Optimum constant estimates | |
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| Mean-square estimation using random variables | |
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| Linear mean-square estimation | |
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| The Stieltjes Integral | |
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| Part V Summary | |
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| Further Topics in Random Variables | |
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| Part VI Introduction | |
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| Complex Random Variables | |
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| The Characteristic Function | |
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| Characteristic Function of a Transformed Random Variable | |
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| Characteristic Function of a Multidimensional Random Variable | |
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| The Generating Function | |
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| Several Jointly Gaussian Random Variables | |
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| Spherically Symmetric Vector Random Variables | |
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| Entropy Associated with Random Variables | |
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| Discrete random variables | |
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| Absolutely continuous random variables | |
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| Copulas | |
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| Sequences of Random Variables | |
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| Preliminaries | |
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| Simple gambling schemes | |
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| Operations on sequences | |
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| Convergent Sequences and Laws of Large Numbers | |
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| Convergence of sequences | |
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| Laws of large numbers | |
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| Connection with statistical regularity | |
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| Convergence of Probability Distributions and the Central Limit Theorem | |
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| Part VI Summary | |
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| Appendices | |
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| Answers to Queries | |
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| Table of the Gaussian Integral | |
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| Part I Problems | |
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| Part II Problems | |
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| Part III Problems | |
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| Part IV Problems | |
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| Part V Problems | |
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| Part VI Problems | |
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| Notation and Abbreviations | |
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| References | |
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| Subject Index | |