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Chapter 1 INTRODUCTION 1

1．1 Introduction 1

1．2 Fundamental problems of probability 3

1．3 Probabilities and sets 4

Chapter 2 PROBABILITY—THE DISCRETE CASE 7

2．1 Discrete sample spaces 7

2．1．1 Subsets and events 9

2．1．2 Operations on sets 10

2．1．3 The algebra of sets 13

2．2 Some combinatorial theory 17

2．2．1 Permutations 20

2．2．2 Combinations 22

2．2．3 Binomial coefficients 23

2．3 Probability 32

2．3．1 The postulates of probability 34

2．3．2 Some elementary theorems of probability 36

2．4 Conditional probability 43

2．5 Some further theorems of probability 52

Chapter 3 PROBABILITY DISTRIBUTIONS 61

3．1 Random variables 61

3．2 Special probability distributions 66

3．2．1 The binomial distribution 66

3．2．2 The hypergeometric distribution 70

3．2．3 The Poisson distribution 72

3．2．4 Some applications 78

3．3 Multivariate probability distributions 81

3．3．1 The multinomial distribution 85

Chapter 4 MATHEMATICAL EXPECTATION:DISCRETE RANDOM VARIABLES 90

4．1 Mathematical expectation 90

4．2 Moments 94

4．2．1 Chebyshev＇s theorem 96

4．3 Moments of special probability distributions 99

4．3．1 Moments of the binomial distribution 99

4．3．2 Moments of the hypergeometric distribution 102

4．3．3 Moments of the Poisson distribution 103

4．4 Moment generating functions 106

4．4．1 The moment generating function of the binomial distribution 108

4．4．2 The moment generating function of the Poisson distribution 108

4．4．3 Moment generating functions and limiting distributions 109

4．5 Product moments 112

4．6 Mathematical expectation and decision making 114

Chapter 5 PROBABILITY DENSITIES 117

5．1 Introduction 117

5．2 Probability densities and distribution functions 117

5．3 Special probability densities 125

5．3．1 The uniform distribution 126

5．3．2 The exponential distribution 126

5．3．3 The gamma distribution 127

5．3．4 The normal distribution 128

5．3．5 Some applications 131

5．4 Change of variable 132

5．5 Multivariate probability densities 137

Chapter 6 MATHEMATICAL EXPECTATION:CONTINUOUS RANDOM VARIABLES 143

6．1 Mathematical expectation 143

6．2 Moments 144

6．3 Moments of special probability densities 145

6．3．1 Moments of the uniform distribution 146

6．3．2 Moments of the gamma distribution 146

6．3．3 Moments of the normal distribution 147

6．4 Moment generating functions 152

6．4．1 Some properties of moment generating functions 153

6．4．2 Moment generating functions of special distributions 154

6．4．3 Moment generating functions and limiting distributions 157

6．5 Product moments 160

6．6 Mathematical expectation and decision making 161

Chapter 7 SUMS OF RANDOM VARIABLES 164

7．1 Introduction 164

7．2 Sums of random variables—convolutions 166

7．3 Sums of random variables—moment generating functions 171

7．4 Moments of linear combinations of random variables 173

7．4．1 The distribution of the mean 175

7．4．2 Differences between means and differences between proportions 179

7．4．3 Sampling from finite populations 181

7．4．4 The distribution of rank sums 183

7．5 The central limit theorem 185

Chapter 8 SAMPLING DISTRIBUTIONS 189

8．1 Introduction 189

8．2 Sampling from normal populations 190

8．2．1 The distribution of X 191

8．2．2 The chi-square distribution and the distribution of S2 193

8．2．3 The F distribution 199

8．2．4 The t distribution 201

8．3 Sampling distributions of order statistics 204

Chapter 9 POINT ESTIMATION 209

9．1 Statistical inference and decision theory 209

9．2 Point estimation and interval estimation 214

9．3 Properties of point estimators 215

9．3．1 Unbiased estimators 215

9．3．2 Consistency 217

9．3．3 Relative efficiency 218

9．3．4 Sufficiency 219

9．4 Methods of point estimation 222

9．4．1 The method of moments 222

9．4．2 The method of maximum likelihood 223

Chapter 10 INTERVAL ESTIMATION 227

10．1 Confidence intervals 227

10．2 Confidence intervals for the mean 230

10．3 Confidence intervals for proportions 232

10．4 Confidence intervals for variances 234

Chapter 11 TESTS OF HYPOTHESES:THEORY 237

11．1 Introduction 237

11．2 Simple hypotheses 238

11．2．1 Type Ⅰ and Type Ⅱ errors 239

11．2．2 The Neyman-Pearson lemma 240

11．3 Composite hypotheses 246

11．3．1 The power function of a test 247

11．3．2 Likelihood ratio tests 251

Chapter 12 TESTS OF HYPOTHESES:APPLICATIONS 259

12．1 Introduction 259

12．2 Tests concerning means 261

12．2．1 One-sample tests concerning means 262

12．2．2 Differences between means 266

12．3 Tests concerning variances 271

12．4 Tests based on count data 274

12．4．1 Tests concerning proportions 275

12．4．2 Differences among k proportions 276

12．4．3 Contingency tables 282

12．4．4 Tests of goodness of fit 285

12．5 Nonparamentric tests 289

12．5．1 The sign test 289

12．5．2 Tests based on rank sums 290

Chapter 13 REGRESSION AND CORRELATION 295

13．1 The problem of regression 295

13．2 Regression curves 296

13．3 Linear regression 299

13．4 The bivariate normal distribution 303

13．4．1 Some sampling theory:correlation analysis 308

13．4．2 Some sampling theory:regression analysis 314

13．5 The method of least squares 321

Chapter 14 INTRODUCTION TO ANALYSIS OF VARIANCE 328

14．1 Introduction 328

14．2 One-way analysis of variance 331

14．3 Two-way analysis of variance 338

14．4 Some further considerations 349

APPENDIX SUMS AND PRODUCTS 351

STATISTICAL TABLES 353

ANSWERS 372

INDEX 385