Statistics for Physical Sciences

by Martin

• ISBN13: 9780123877604
• ISBN10: 0123877601
• Format: Hardcover
• Copyright: 2012-01-19
• Publisher: Elsevier Science

Summary

Statistical Methods for the Physical Sciences is an informal, relatively short, but systematic, guide to the more commonly used ideas and techniques in statistical analysis, as used in physical sciences, together with explanations of their origins. It steers a path between the extremes of a recipe of methods with a collection of useful formulas, and a full mathematical account of statistics, while at the same time developing the subject in a logical way. The book can be read in its entirety by anyone with a basic exposure to mathematics at the level of a first-year undergraduate student of physical science and should be useful for practising physical scientists, plus undergraduate and postgraduate students in these fields. - Problems at the end of each chapter - All the chapters contain worked examples - Collection of useful formulas in order to give a detailed account of mathematical statistics

Table of Contents

Preface	p. ix
Statistics, Experiments, and Data	p. 1
Experiments and Observations	p. 2
Displaying Data	p. 4
Summarizing Data Numerically	p. 7
Measures of Location	p. 8
Measures of Spread	p. 9
More than One Variable	p. 12
Large Samples	p. 15
Experimental Errors	p. 17
Problems 1	p. 19
Probability	p. 21
Axioms of Probability	p. 21
Calculus of Probabilities	p. 23
The Meaning of Probability	p. 27
Frequency Interpretation	p. 27
Subjective Interpretation	p. 29
Problems 2	p. 32
Probability Distributions I: Basic Concepts	p. 35
Random Variables	p. 35
Single Variates	p. 36
Probability Distributions	p. 36
Expectation Values	p. 40
Moment Generating, and Characteristic Functions	p. 42
Several Variates	p. 45
Joint Probability Distributions	p. 45
Marginal and Conditional Distributions	p. 45
Moments and Expectation Values	p. 49
Functions of a Random Variable	p. 51
Problems 3	p. 55
Probability Distributions II: Examples	p. 57
Uniform	p. 57
Univariate Normal (Gaussian)	p. 59
Multivariate Normal	p. 63
Bivariate Normal	p. 65
Exponential	p. 66
Cauchy	p. 68
Binomial	p. 69
Multinomial	p. 74
Poisson	p. 75
Problems 4	p. 80
Sampling and Estimation	p. 83
Random Samples and Estimators	p. 83
Sampling Distributions	p. 84
Properties of Point Estimators	p. 86
Estimators for the Mean, Variance, and Covariance	p. 90
Laws of Large Numbers and the Central Limit Theorem	p. 93
Experimental Errors	p. 97
Propagation of Enors	p. 99
Problems 5	p. 103
Sampling Distributions Associated with the Normal Distribution	p. 105
Chi-Squared Distribution	p. 105
Student's t Distribution	p. 111
F Distribution	p. 116
Relations Between ¿2, t, and F Distributions	p. 119
Problems 6	p. 121
Parameter Estimation I: Maximum Likelihood and Minimum Variance	p. 123
Estimation of a Single Parameter	p. 123
Variance of an Estimator	p. 128
Approximate methods	p. 130
Simultaneous Estimation of Several Parameters	p. 133
Minimum Variance	p. 136
Parameter Estimation	p. 136
Minimum Variance Bound	p. 137
Problems 7	p. 140
Parameter Estimation II: Least-Squares and Other Methods	p. 143
Unconstrained Linear Least Squares	p. 143
General Solution for the Parameters	p. 145
Errors on the Parameter Estimates	p. 149
Quality of the Fit	p. 151
Orthogonal Polynomials	p. 152
Fitting a Straight Line	p. 154
Combining Experiments	p. 158
Linear Least Squares with Constraints	p. 159
Nonlinear Least Squares	p. 162
Other Methods	p. 163
Minimum Chi-Square	p. 163
Method of Moments	p. 165
Bayes' Estimators	p. 167
Problems 8	p. 171
Interval Estimation	p. 173
Confidence Intervals: Basic Ideas	p. 174
Confidence Intervals: General Method	p. 177
Normal Distribution	p. 179
Confidence Intervals for the Mean	p. 180
Confidence Intervals for the Variance	p. 182
Confidence Regions for the Mean and Variance	p. 183
Poisson Distribution	p. 184
Large Samples	p. 186
Confidence Intervals Near Boundaries	p. 187
Bayesian Confidence Intervals	p. 189
Problems 9	p. 190
Hypothesis Testing I: Parameters	p. 193
Statistical Hypotheses	p. 194
General Hypotheses: Likelihood Ratios	p. 198
Simple Hypothesis: One Simple Alternative	p. 198
Composite Hypotheses	p. 201
Normal Distribution	p. 204
Basic Ideas	p. 204
Specific Tests	p. 206
Other Distributions	p. 214
Analysis of Variance	p. 215
Problems 10	p. 218
Hypothesis Testing II: Other Tests	p. 221
Goodness-of-Fit Tests	p. 221
Discrete Distributions	p. 222
Continuous Distributions	p. 225
Linear Hypotheses	p. 228
Tests for Independence	p. 231
Nonparametric Tests	p. 233
Sign Test	p. 233
Signed-Rank Test	p. 234
Rank-Sum Test	p. 236
Runs Test	p. 237
Rank Correlation Coefficient	p. 239
Problems 11	p. 241
Miscellaneous Mathematics	p. 243
Matrix Algebra	p. 243
Classical Theory of Minima	p. 247
Optimization of Nonlinear Functions	p. 249
General Principles	p. 249
Unconstrained Minimization of Functions of One variable	p. 252
Unconstrained Minimization of Multivariable Functions	p. 253
Direct Search Methods	p. 253
Gradient Methods	p. 254
Constrained Optimization	p. 255
Statistical Tables	p. 257
Normal Distribution	p. 257
Binomial Distribution	p. 259
Poisson Distribution	p. 266
Chi-squared Distribution	p. 273
Student's t Distribution	p. 275
F Distribution	p. 277
Signed-Rank Test	p. 283
Rank-Sum Test	p. 284
Runs Test	p. 285
Rank Correlation Coefficient	p. 286
Answers to Odd-Numbered Problems	p. 287
Bibliography	p. 293
Index	p. 295
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