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Foreword | |

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Preface | |

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Survey of probability theory | |

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Introduction | |

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Experiments, Sample Spaces, and Probability | |

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Experiments and Sample Spaces | |

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Set Theory | |

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Events and Probability | |

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Conditional Probability | |

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Binomial Coefficients | |

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Exercises | |

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Random Variables, Random Vectors, and Distribution Functions | |

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Random Variables and Their Distributions | |

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Multivariate Distributions | |

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Sums and Integrals | |

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Marginal Distributions and Independence | |

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Vectors and Matrices | |

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Expectations, Moments, and Characteristic Functions | |

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Transformations of Random Variables | |

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Conditional Distributions | |

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Exercises | |

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Some Special Univariate Distributions | |

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Introduction | |

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The Bernoulli Distribution | |

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The Binomial Distribution | |

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The Poisson Distribution | |

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The Negative Binomial Distribution | |

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The Hypergeometric Distribution | |

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The Normal Distribution | |

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The Gamma Distribution | |

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The Beta Distribution | |

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The Uniform Distribution | |

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The Pareto Distribution | |

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The t Distribution | |

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The F Distribution | |

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Exercises | |

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Some Special Multivariate Distributions | |

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Introduction | |

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The Multinomial Distribution | |

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The Dirichlet Distribution | |

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The Multivariate Normal Distribution | |

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The Wishart Distribution | |

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The Multivariate t Distribution | |

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The Bilateral Bivariate Pareto Distribution | |

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Exercises | |

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Subjective probability and utility | |

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Subjective Probability | |

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Introduction | |

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Relative Likelihood | |

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The Auxiliary Experiment | |

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Construction of the Probability Distribution | |

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Verification of the Properties of a Probability Distribution | |

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Conditional Likelihoods | |

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Exercises | |

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Utility | |

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Preferences among Rewards | |

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Preferences among Probability Distributions | |

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The Definition of a Utility Function | |

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Some Properties of Utility Functions | |

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The Utility of Monetary Rewards | |

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Convex and Concave Utility Functions | |

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The Axiomatic Development of Utility | |

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Construction of the Utility Function | |

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Verification of the Properties of a Utility Function | |

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Extension of the Properties of a Utility Function to the Class P[subscript E] | |

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Exercises | |

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Statistical decision problems | |

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Decision Problems | |

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Elements of a Decision Problem | |

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Bayes Risk and Bayes Decisions | |

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Nonnegative Loss Functions | |

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Concavity of the Bayes Risk | |

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Randomization and Mixed Decisions | |

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Convex Sets | |

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Decision Problems in Which [similar]2 and D Are Finite | |

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Decision Problems with Observations | |

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Construction of Bayes Decision Functions | |

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The Cost of Observation | |

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Statistical Decision Problems in Which Both [Omega] and D Contain Two Points | |

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Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage | |

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Exercises | |

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Conjugate Prior Distributions | |

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Sufficient Statistics | |

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Conjugate Families of Distributions | |

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Construction of the Conjugate Family | |

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Conjugate Families for Samples from Various Standard Distributions | |

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Conjugate Families for Samples from a Normal Distribution | |

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Sampling from a Normal Distribution with Unknown Mean and Unknown Precision | |

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Sampling from a Uniform Distribution | |

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A Conjugate Family for Multinomial Observations | |

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Conjugate Families for Samples from a Multivariate Normal Distribution | |

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Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix | |

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The Marginal Distribution of the Mean Vector | |

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The Distribution of a Correlation | |

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Precision Matrices Having an Unknown Factor | |

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Exercises | |

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Limiting Posterior Distributions | |

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Improper Prior Distributions | |

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Improper Prior Distributions for Samples from a Normal Distribution | |

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Improper Prior Distributions for Samples from a Multivariate Normal Distribution | |

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Precise Measurement | |

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Convergence of Posterior Distributions | |

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Supercontinuity | |

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Solutions of the Likelihood Equation | |

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Convergence of Supercontinuous Functions | |

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Limiting Properties of the Likelihood Function | |

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Normal Approximation to the Posterior Distribution | |

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Approximations for Vector Parameters | |

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Posterior Ratios | |

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Exercises | |

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Estimation, Testing Hypotheses, and Linear Statistical Models | |

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Estimation | |

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Quadratic Loss | |

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Loss Proportional to the Absolute Value of the Error | |

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Estimation of a Vector | |

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Problems of Testing Hypotheses | |

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Testing a Simple Hypothesis about the Mean of a Normal Distribution | |

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Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown | |

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Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value | |

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Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value | |

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Linear Models | |

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Testing Hypotheses in Linear Models | |

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Investigating the Hypothesis That Certain Regression Coefficients Vanish | |

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One-way Analysis of Variance | |

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Exercises | |

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Sequential decisions | |

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Sequential Sampling | |

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Gains from Sequential Sampling | |

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Sequential Decision Procedures | |

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The Risk of a Sequential Decision Procedure | |

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Backward Induction | |

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Optimal Bounded Sequential Decision Procedures | |

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Illustrative Examples | |

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Unbounded Sequential Decision Procedures | |

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Regular Sequential Decision Procedures | |

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Existence of an Optimal Procedure | |

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Approximating an Optimal Procedure by Bounded Procedures | |

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Regions for Continuing or Terminating Sampling | |

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The Functional Equation | |

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Approximations and Bounds for the Bayes Risk | |

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The Sequential Probability-ratio Test | |

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Characteristics of Sequential Probability-ratio Tests | |

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Approximating the Expected Number of Observations | |

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Exercises | |

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Optimal Stopping | |

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Introduction | |

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The Statistician's Reward | |

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Choice of the Utility Function | |

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Sampling without Recall | |

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Further Problems of Sampling with Recall and Sampling without Recall | |

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Sampling without Recall from a Normal Distribution with Unknown Mean | |

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Sampling with Recall from a Normal Distribution with Unknown Mean | |

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Existence of Optimal Stopping Rules | |

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Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall | |

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Martingales | |

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Stopping Rules for Martingales | |

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Uniformly Integrable Sequences of Random Variables | |

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Martingales Formed from Sums and Products of Random Variables | |

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Regular Supermartingales | |

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Supermartingales and General Problems of Optimal Stopping | |

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Markov Processes | |

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Stationary Stopping Rules for Markov Processes | |

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Entrance-fee Problems | |

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The Functional Equation for a Markov Process | |

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Exercises | |

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Sequential Choice of Experiments | |

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Introduction | |

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Markovian Decision Processes with a Finite Number of Stages | |

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Markovian Decision Processes with an Infinite Number of Stages | |

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Some Betting Problems | |

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Two-armed-bandit Problems | |

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Two-armed-bandit Problems When the Value of One Parameter Is Known | |

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Two-armed-bandit Problems When the Parameters Are Dependent | |

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Inventory Problems | |

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Inventory Problems with an Infinite Number of Stages | |

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Control Problems | |

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Optimal Control When the Process Cannot Be Observed without Error | |

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Multidimensional Control Problems | |

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Control Problems with Actuation Errors | |

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Search Problems | |

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Search Problems with Equal Costs | |

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Uncertainty Functions and Statistical Decision Problems | |

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Sufficient Experiments | |

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Examples of Sufficient Experiments | |

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Exercises | |

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References | |

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Supplementary Bibliography | |

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Name Index | |

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Subject Index | |