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Chapter 1 INTRODUCTION1

1．1 Introduction1

1．2 Fundamental problems of probability3

1．3 Probabilities and sets4

Chapter 2 PROBABILITY—THE DISCRETE CASE7

2．1 Discrete sample spaces7

2．1．1 Subsets and events9

2．1．2 Operations on sets10

2．1．3 The algebra of sets13

2．2 Some combinatorial theory17

2．2．1 Permutations20

2．2．2 Combinations22

2．2．3 Binomial coefficients23

2．3 Probability32

2．3．1 The postulates of probability34

2．3．2 Some elementary theorems of probability36

2．4 Conditional probability43

2．5 Some further theorems of probability52

Chapter 3 PROBABILITY DISTRIBUTIONS61

3．1 Random variables61

3．2 Special probability distributions66

3．2．1 The binomial distribution66

3．2．2 The hypergeometric distribution70

3．2．3 The Poisson distribution72

3．2．4 Some applications78

3．3 Multivariate probability distributions81

3．3．1 The multinomial distribution85

Chapter 4 MATHEMATICAL EXPECTATION:DISCRETE RANDOM VARIABLES90

4．1 Mathematical expectation90

4．2 Moments94

4．2．1 Chebyshev＇s theorem96

4．3 Moments of special probability distributions99

4．3．1 Moments of the binomial distribution99

4．3．2 Moments of the hypergeometric distribution102

4．3．3 Moments of the Poisson distribution103

4．4 Moment generating functions106

4．4．1 The moment generating function of the binomial distribution108

4．4．2 The moment generating function of the Poisson distribution108

4．4．3 Moment generating functions and limiting distributions109

4．5 Product moments112

4．6 Mathematical expectation and decision making114

Chapter 5 PROBABILITY DENSITIES117

5．1 Introduction117

5．2 Probability densities and distribution functions117

5．3 Special probability densities125

5．3．1 The uniform distribution126

5．3．2 The exponential distribution126

5．3．3 The gamma distribution127

5．3．4 The normal distribution128

5．3．5 Some applications131

5．4 Change of variable132

5．5 Multivariate probability densities137

Chapter 6 MATHEMATICAL EXPECTATION:CONTINUOUS RANDOM VARIABLES143

6．1 Mathematical expectation143

6．2 Moments144

6．3 Moments of special probability densities145

6．3．1 Moments of the uniform distribution146

6．3．2 Moments of the gamma distribution146

6．3．3 Moments of the normal distribution147

6．4 Moment generating functions152

6．4．1 Some properties of moment generating functions153

6．4．2 Moment generating functions of special distributions154

6．4．3 Moment generating functions and limiting distributions157

6．5 Product moments160

6．6 Mathematical expectation and decision making161

Chapter 7 SUMS OF RANDOM VARIABLES164

7．1 Introduction164

7．2 Sums of random variables—convolutions166

7．3 Sums of random variables—moment generating functions171

7．4 Moments of linear combinations of random variables173

7．4．1 The distribution of the mean175

7．4．2 Differences between means and differences between proportions179

7．4．3 Sampling from finite populations181

7．4．4 The distribution of rank sums183

7．5 The central limit theorem185

Chapter 8 SAMPLING DISTRIBUTIONS189

8．1 Introduction189

8．2 Sampling from normal populations190

8．2．1 The distribution of X191

8．2．2 The chi-square distribution and the distribution of S2193

8．2．3 The F distribution199

8．2．4 The t distribution201

8．3 Sampling distributions of order statistics204

Chapter 9 POINT ESTIMATION209

9．1 Statistical inference and decision theory209

9．2 Point estimation and interval estimation214

9．3 Properties of point estimators215

9．3．1 Unbiased estimators215

9．3．2 Consistency217

9．3．3 Relative efficiency218

9．3．4 Sufficiency219

9．4 Methods of point estimation222

9．4．1 The method of moments222

9．4．2 The method of maximum likelihood223

Chapter 10 INTERVAL ESTIMATION227

10．1 Confidence intervals227

10．2 Confidence intervals for the mean230

10．3 Confidence intervals for proportions232

10．4 Confidence intervals for variances234

Chapter 11 TESTS OF HYPOTHESES:THEORY237

11．1 Introduction237

11．2 Simple hypotheses238

11．2．1 Type Ⅰ and Type Ⅱ errors239

11．2．2 The Neyman-Pearson lemma240

11．3 Composite hypotheses246

11．3．1 The power function of a test247

11．3．2 Likelihood ratio tests251

Chapter 12 TESTS OF HYPOTHESES:APPLICATIONS259

12．1 Introduction259

12．2 Tests concerning means261

12．2．1 One-sample tests concerning means262

12．2．2 Differences between means266

12．3 Tests concerning variances271

12．4 Tests based on count data274

12．4．1 Tests concerning proportions275

12．4．2 Differences among k proportions276

12．4．3 Contingency tables282

12．4．4 Tests of goodness of fit285

12．5 Nonparamentric tests289

12．5．1 The sign test289

12．5．2 Tests based on rank sums290

Chapter 13 REGRESSION AND CORRELATION295

13．1 The problem of regression295

13．2 Regression curves296

13．3 Linear regression299

13．4 The bivariate normal distribution303

13．4．1 Some sampling theory:correlation analysis308

13．4．2 Some sampling theory:regression analysis314

13．5 The method of least squares321

Chapter 14 INTRODUCTION TO ANALYSIS OF VARIANCE328

14．1 Introduction328

14．2 One-way analysis of variance331

14．3 Two-way analysis of variance338

14．4 Some further considerations349

APPENDIX SUMS AND PRODUCTS351

STATISTICAL TABLES353

ANSWERS372

INDEX385