Preface Chapter1 Probabilityand Distributions 1.1 Probability 1.1.1 Permutation， combination and binomial coefficients 1.1.2 Sample space 1.1.3 Events 1.1.4 Propertiesof probability 1.2 Conditional Probability 1.3 Bayes Theorem 1.4 ProbabilityDistributions 1.5 Bivariate Distributions 1.5.1 Joint distribution 1.5.2 Marginal and conditional distributions 1.5.3 Independencyoftwo randomvariables 1.6 Expectation，Variance and Moments 1.6.1 Moments 1.6.2 Some probabilityinequalities 1.6.3 Conditional expectation 1.6.4 Compound randomvariables 1.6.5 Calculation of (conditional) probabilityvia (conditional) expectation 1.7 Moment GeneratingFunction 1.8 Beta and Gamma Distributions 1.8.1 Beta distribution 1.8.2 Gamma distribution 1.9 Bivariate Normal Distribution 1.9.1 Univariate normal distribution 1.9.2 Correlation coefficient 1.9.3 Joint density 1.9.4 Stochastic representation of random variables or random vectors Contents 1.10 Inverse Bayes Formulae 1.10.1 Three inverse Bayes formulae 1.10.2 Understanding the IBF 1.10.3 Two examples 1.11 Categorical Distribution 1.12 Zero-inflatedPoisson Distribution Exercise Chapter2 Sampling Distributions 2.1 Distribution of the Function of RandomVariables 2.1.1 Cumulative distribution function technique 2.1.2 Transformation technique 2.1.3 Momentgenerating function technique 2.2 Statistics， Sample Mean and SampleVariance 2.2.1 Distributionofthe sample mean 2.2.2 Distributionofthe samplevariance 2.3 The and Distributions 2.3.1 The distribution 2.3.2 The distribution 2.4 Order Statistics 2.4.1 Distributionofa single order statistic 2.4.2 Joint distributionof more order statistics
2.5 Limit Theorems 2.5.1 Convergencyofa sequenceof distribution functions 2.5.2 Convergencein probability 2.5.3 Relationshipof four classesof convergency 2.5.4 Lawof largenumber 2.5.5 Central limit theorem 2.6 Some Challenging Questions Exercise Chapter3 Point Estimation 3.1 Maximum LikelihoodEstimator 3.1.1 Pointestimator andpointestimate 3.1.2 Joint densityand likelihoodfunction 3.1.3 Maximum likelihoodestimate and maximum likelihood estimator 3.1.4 Theinvariance propertyof MLE Contents vii 3.2 Moment Estimator 3.3 Bayesian Estimator 3.4 Propertiesof Estimators 3.4.1 Unbiasedness 3.4.2 Efficiency 3.4.3 Sufficiency 3.4.4 Completeness 3.5 Limiting Properties of MLE 3.6 Some Challenging Questions Exercise Chapter4 Confidence Interval Estimation 4.1 Introduction 4.2 The ConfidenceIntervalof Normal Mean 4.2.1 Thevarianceisknown 4.2.2 Thevarianceis unknown 4.3 The Confidence Interval of the Difference of Two Normal Means 4.4 The ConfidenceInterval of Normal Variance 4.4.1 The mean is known 4.4.2 The meanis unknown 4.5 The Confidence Interval of the Ratio of Two Normal Variances 4.6 Large-Sample ConfidenceIntervals 4.7 The Shortest ConfidenceInterval Exercise Chapter5 Hypothesis Testing 5.1 Introduction 5.1.1 Several basic notions 5.1.2 TypeIerror andTypeII error 5.1.3 Power function 5.2 The Neyman–Pearson Lemma 5.2.1 Simplenullhypothesisversus simple alternative 5.2.2 Compositehypotheses 5.3 LikelihoodRatioTest 5.3.1 Likelihoodratio statistic 5.3.2 Likelihoodratio test 5.4 Testson Normal Means 5.4.1 One–sample normal test whenvarianceisknown
5.4.2 One–sample test 5.4.3 Two–samplet test 5.5 GoodnessofFitTest 5.5.1 Introduction 5.5.2 Thechi-square testfor totallyknown distribution 5.5.3 The chi-square test for known distribution family with unknown parameters Exercise Chapter6 Critical Regions and p-values for Skew Null Distributions 6.1 One–sample Chi-square Test on Normal Variance 6.2 Two–sampleF Test on Normal Variances Appendix A Basic Statistical Distributions A.Discrete Distributions A.Continuous Distributions Appendix B AUnified Expectation Technique B.Continuous RandomVariables B.Discrete RandomVariables Appendix C The Newton–Raphson and Fisher Scoring Algorithms C.Newton』s Method fo rRoot Finding C.Newton』s Method for CalculatingMLE C.The Newton–Raphson Algorithm for High-dimensional Cases C.The Fisher Scoring Algorithm List of Figures List ofTables List ofAcronyms List of Symbols References Subject Index
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