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[资源] Statistics for Physical Sciences. An Introduction

Contents
Preface ix
1. Statistics, Experiments, and Data 1
1.1. Experiments and Observations 2
1.2. Displaying Data 4
1.3. Summarizing Data Numerically 7
1.3.1. Measures of Location 8
1.3.2. Measures of Spread 9
1.3.3. More than One Variable 12
1.4. Large Samples 15
1.5. Experimental Errors 17
Problems 1 19
2. Probability 21
2.1. Axioms of Probability 21
2.2. Calculus of Probabilities 23
2.3. The Meaning of Probability 27
2.3.1. Frequency Interpretation 27
2.3.2. Subjective Interpretation 29
Problems 2 32
3. Probability Distributions I:
Basic Concepts 35
3.1. Random Variables 35
3.2. Single Variates 36
3.2.1. Probability Distributions 36
3.2.2. Expectation Values 40
3.2.3. Moment Generating, and Characteristic
Functions 42
3.3. Several Variates 45
3.3.1. Joint Probability Distributions 45
3.3.2. Marginal and Conditional
Distributions 45
3.3.3. Moments and Expectation Values 49
3.4. Functions of a Random Variable 51
Problems 3 55
4. Probability Distributions II:
Examples 57
4.1. Uniform 57
4.2. Univariate Normal (Gaussian) 59
4.3. Multivariate Normal 63
4.3.1. Bivariate Normal 65
4.4. Exponential 66
4.5. Cauchy 68
4.6. Binomial 69
4.7. Multinomial 74
4.8. Poisson 75
Problems 4 80
5. Sampling and Estimation 83
5.1. Random Samples and Estimators 83
5.1.1. Sampling Distributions 84
5.1.2. Properties of Point Estimators 86
5.2. Estimators for the Mean, Variance, and
Covariance 90
5.3. Laws of Large Numbers and the Central Limit
Theorem 93
5.4. Experimental Errors 97
5.4.1. Propagation of Errors 99
Problems 5 103
6. Sampling Distributions Associated with
the Normal Distribution 105
6.1. Chi-Squared Distribution 105
6.2. Student’s t Distribution 111
6.3. F Distribution 116
6.4. Relations Between c 2 , t, and F
Distributions 119
Problems 6 121
v
7. Parameter Estimation I: Maximum
Likelihood and Minimum Variance 123
7.1. Estimation of a Single Parameter 123
7.2. Variance of an Estimator 128
7.2.1. Approximate methods 130
7.3. Simultaneous Estimation of Several
Parameters 133
7.4. Minimum Variance 136
7.4.1. Parameter Estimation 136
7.4.2. Minimum Variance Bound 137
Problems 7 140
8. Parameter Estimation II: Least-Squares
and Other Methods 143
8.1. Unconstrained Linear Least Squares 143
8.1.1. General Solution for the Parameters 145
8.1.2. Errors on the Parameter Estimates 149
8.1.3. Quality of the Fit 151
8.1.4. Orthogonal Polynomials 152
8.1.5. Fitting a Straight Line 154
8.1.6. Combining Experiments 158
8.2. Linear Least Squares with Constraints 159
8.3. Nonlinear Least Squares 162
8.4. Other Methods 163
8.4.1. Minimum Chi-Square 163
8.4.2. Method of Moments 165
8.4.3. Bayes’ Estimators 167
Problems 8 171
9. Interval Estimation 173
9.1. Confidence Intervals: Basic Ideas 174
9.2. Confidence Intervals: General Method 177
9.3. Normal Distribution 179
9.3.1. Confidence Intervals for the Mean 180
9.3.2. Confidence Intervals for the
Variance 182
9.3.3. Confidence Regions for the Mean and
Variance 183
9.4. Poisson Distribution 184
9.5. Large Samples 186
9.6. Confidence Intervals Near Boundaries 187
9.7. Bayesian Confidence Intervals 189
Problems 9 190
10. Hypothesis Testing I: Parameters 193
10.1. Statistical Hypotheses 194
10.2. General Hypotheses: Likelihood Ratios 198
10.2.1. Simple Hypothesis: One Simple
Alternative 198
10.2.2. Composite Hypotheses 201
10.3. Normal Distribution 204
10.3.1. Basic Ideas 204
10.3.2. Specific Tests 206
10.4. Other Distributions 214
10.5. Analysis of Variance 215
Problems 10 218
11. Hypothesis Testing II: Other
Tests 221
11.1. Goodness-of-Fit Tests 221
11.1.1. Discrete Distributions 222
11.1.2. Continuous Distributions 225
11.1.3. Linear Hypotheses 228
11.2. Tests for Independence 231
11.3. Nonparametric Tests 233
11.3.1. Sign Test 233
11.3.2. Signed-Rank Test 234
11.3.3. Rank-Sum Test 236
11.3.4. Runs Test 237
11.3.5. Rank Correlation Coefficient 239
Problems 11 241
Appendix A. Miscellaneous
Mathematics 243
A.1. Matrix Algebra 243
A.2. Classical Theory of Minima 247
Appendix B. Optimization of Nonlinear
Functions 249
B.1. General Principles 249
B.2. Unconstrained Minimization of Functions of
One variable 252
B.3. Unconstrained Minimization of Multivariable
Functions 253
B.3.1. Direct Search Methods 253
B.3.2. Gradient Methods 254
B.4. Constrained Optimization 255
CONTENTS vi
Appendix C. Statistical Tables 257
C.1. Normal Distribution 257
C.2. Binomial Distribution 259
C.3. Poisson Distribution 266
C.4. Chi-squared Distribution 273
C.5. Student’s t Distribution 275
C.6. F Distribution 277
C.7. Signed-Rank Test 283
C.8. Rank-Sum Test 284
C.9. Runs Test 285
C.10. Rank Correlation Coefficient 286
Appendix D. Answers to Odd-Numbered
Problems 287
Bibliography 293
Index 295

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2016-09-13 19:22:30, 3.89 M