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 x2, t, and F Distributions .......... 119
Problems 6 ............................................ 121
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.l Matrix Algebra ........................................ 243
A.2 Classical Theory of Minima ............................ 247
Appendix B. Optimization of Nonlinear Functions ............... 249
B.l 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
Appendix C. Statistical Tables ................................ 257
С.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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