Contents
Preface
Chapter1 Probabilityand Distributions 1
1.1 Probability 1
1.1.1 Permutation， combination and binomial coefficients 1
1.1.2 Sample space 3
1.1.3 Events 4
1.1.4 Propertiesof probability 5
1.2 Conditional Probability 7
1.3 Bayes Theorem 9
1.4 ProbabilityDistributions 10
1.5 Bivariate Distributions 13
1.5.1 Joint distribution 13
1.5.2 Marginal and conditional distributions 14
1.5.3 Independencyoftwo randomvariables 14
1.6 Expectation，Variance and Moments 16
1.6.1 Moments 16
1.6.2 Some probabilityinequalities 18
1.6.3 Conditional expectation 21
1.6.4 Compound randomvariables 23
1.6.5 Calculation of conditional probabilityvia conditional expectation 23
1.7 Moment GeneratingFunction 24
1.8 Beta and Gamma Distributions 27
1.8.1 Beta distribution 27
1.8.2 Gamma distribution 29
1.9 Bivariate Normal Distribution 32
1.9.1 Univariate normal distribution 32
1.9.2 Correlation coefficient 34
1.9.3 Joint density 34
1.9.4 Stochastic representation of random variables or random vectors 38
Contents 1.10 Inverse Bayes Formulae 40
1.10.1 Three inverse Bayes formulae 40
1.10.2 Understanding the IBF 43
1.10.3 Two examples 45
1.11 Categorical Distribution 47
1.12 Zero-inflatedPoisson Distribution 49
Exercise 1 53
Chapter2 Sampling Distributions 57
2.1 Distribution of the Function of RandomVariables 57
2.1.1 Cumulative distribution function technique 57
2.1.2 Transformation technique 62
2.1.3 Momentgenerating function technique 71
2.2 Statistics， Sample Mean and SampleVariance 73
2.2.1 Distributionofthe sample mean 73
2.2.2 Distributionofthe samplevariance 74
2.3 The and Distributions 76
2.3.1 The distribution 76
2.3.2 The distribution 78
2.4 Order Statistics 81
2.4.1 Distributionofa single order statistic 81
2.4.2 Joint distributionof more order statistics 84
2.5 Limit Theorems 86
2.5.1 Convergencyofa sequenceof distribution functions 86
2.5.2 Convergencein probability 91
2.5.3 Relationshipof four classesof convergency 92
2.5.4 Lawof largenumber 94
2.5.5 Central limit theorem 94
2.6 Some Challenging Questions 96
Exercise 2 99
Chapter3 Point Estimation 102
3.1 Maximum LikelihoodEstimator 102
3.1.1 Pointestimator andpointestimate 102
3.1.2 Joint densityand likelihoodfunction 104
3.1.3 Maximum likelihoodestimate and maximum likelihood estimator 105
3.1.4 Theinvariance propertyof MLE 115
Contents vii 3.2 Moment Estimator 117
3.3 Bayesian Estimator 121
3.4 Propertiesof Estimators 125
3.4.1 Unbiasedness 125
3.4.2 Efficiency 126
3.4.3 Sufficiency 138
3.4.4 Completeness 146
3.5 Limiting Properties of MLE 151
3.6 Some Challenging Questions 153
Exercise 3 156
Chapter4 Confidence Interval Estimation 162
4.1 Introduction 162
4.2 The ConfidenceIntervalof Normal Mean 166
4.2.1 Thevarianceisknown 166
4.2.2 Thevarianceis unknown 167
4.3 The Confidence Interval of the Difference of Two Normal Means 169
4.4 The ConfidenceInterval of Normal Variance 171
4.4.1 The mean is known 171
4.4.2 The meanis unknown 172
4.5 The Confidence Interval of the Ratio of Two Normal Variances 172
4.6 Large-Sample ConfidenceIntervals 174
4.7 The Shortest ConfidenceInterval 178
Exercise 4 180
Chapter5 Hypothesis Testing 183
5.1 Introduction 183
5.1.1 Several basic notions 184
5.1.2 TypeIerror andTypeII error 186
5.1.3 Power function 189
5.2 The Neyman–Pearson Lemma 191
5.2.1 Simplenullhypothesisversus simple alternative 192
5.2.2 Compositehypotheses 199
5.3 LikelihoodRatioTest 203
5.3.1 Likelihoodratio statistic 203
5.3.2 Likelihoodratio test 205
5.4 Testson Normal Means 211
5.4.1 One–sample normal test whenvarianceisknown 211
5.4.2 One–sample test 215
5.4.3 Two–samplet test 217
5.5 GoodnessofFitTest 220
5.5.1 Introduction 220
5.5.2 Thechi-square testfor totallyknown distribution 222
5.5.3 The chi-square test for known distribution family with unknown parameters 226
Exercise 5 230
Chapter6 Critical Regions and p-values for Skew Null Distributions 233
6.1 One–sample Chi-square Test on Normal Variance 233
6.2 Two–sampleF Test on Normal Variances 238
Appendix A Basic Statistical Distributions 246
A.Discrete Distributions 246
A.Continuous Distributions 250
Appendix B AUnified Expectation Technique 256
B.Continuous RandomVariables 257
B.Discrete RandomVariables 277
Appendix C The Newton–Raphson and Fisher Scoring Algorithms 289
C.Newton’s Method fo rRoot Finding 289
C.Newton’s Method for CalculatingMLE 294
C.The Newton–Raphson Algorithm for High-dimensional Cases 299
C.The Fisher Scoring Algorithm 304
List of Figures 307
List ofTables 309
List ofAcronyms 310
List of Symbols 311
References 315
Subject Index 317
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