Probability And Statistics In The Physical Sciences

Preface......Page 6
Acknowledgements......Page 8
Contents......Page 9
1.1 Discussion of Randomness, Probability,  and a Priori Probability......Page 14
References......Page 17
2.1 Basic Concepts......Page 18
2.2 Worked Problems......Page 23
2.3 Exercises......Page 25
Reference......Page 26
3.1 Multiple Scattering and the Root N Law......Page 27
3.2 Propagation of Errors; Errors When Changing Variables......Page 30
3.3 Some Useful Inequalities......Page 32
3.4 Worked Problems......Page 34
3.5 Exercises......Page 37
References......Page 38
4.1 Combinatorials......Page 39
4.2 Stirling\'s Approximation for the Factorial Function......Page 41
4.3 Worked Problems......Page 42
4.4 Exercises......Page 43
5.1 Binomial Distribution......Page 44
5.2 Worked Problems......Page 48
5.3 Exercises......Page 51
Reference......Page 52
6.1 The Normal Distribution......Page 53
6.2 The χ2 Distribution......Page 57
6.3 F Distribution......Page 59
6.5 The Uniform Distribution......Page 60
6.6 The Log-Normal Distribution......Page 61
6.7 The Breit–Wigner (Cauchy) Distribution......Page 62
6.8 Beta Distribution......Page 63
6.9.1 Exponential Distribution......Page 64
6.9.2 Double Exponential Distribution......Page 65
6.11 Worked Problems......Page 66
6.12 Exercises......Page 68
Reference......Page 69
7.1 Convolutions and Compound Probability......Page 70
7.2 Generating Functions......Page 71
7.3 Characteristic Functions......Page 75
7.4 Exercises......Page 77
8.1 Using the Distribution Inverse......Page 79
8.2 Method of Composition......Page 80
8.3 Acceptance Rejection Method......Page 82
8.4 Computer Pseudorandom Number Generators......Page 83
8.5 Unusual Application of a Pseudorandom Number String......Page 84
8.6 Worked Problems......Page 86
8.7 Exercises......Page 88
References......Page 90
9.1 Queueing Theory......Page 92
9.2 Markov Chains......Page 94
9.3 Games of Chance......Page 97
9.4 Gambler\'s Ruin......Page 99
References......Page 101
10.1 Two-Dimensional Distributions......Page 102
10.2 Multidimensional Distributions......Page 108
10.3 Exercises......Page 112
11.1 The Lindeberg Criterion......Page 113
11.2 Failures of the Central Limit Theorem......Page 115
11.3 Khintchine\'s Law of the Iterated Logarithm......Page 119
11.4 Worked Problems......Page 120
11.5 Exercises......Page 122
References......Page 123
12.1 Lemma of Fisher......Page 124
12.2 Maximum Likelihood Method......Page 126
12.3 Testing and Comparing Hypotheses......Page 130
References......Page 132
13.1 Least Squares Analysis......Page 133
13.2 Problem with the Use of χ2 if the Number of Events Is Not Fixed......Page 134
13.3 Further Considerations in Fitting Histograms......Page 141
13.4 Estimation of a Correlation Coefficient......Page 142
13.5 Worked Problems......Page 143
13.6 Exercises......Page 145
References......Page 147
14.1 Bayes Theorem......Page 148
14.2 The Problem of A Priori Probability......Page 150
14.3 Confidence Intervals and Their Interpretation......Page 151
14.4 Use of Confidence Intervals for Discrete Distributions......Page 156
14.5 Improving on the Symmetric Tails Confidence Limits......Page 159
14.6 Improvement over Symmetric Tails Confidence Limits for Events …......Page 166
14.7 When Is a Signal Significant?......Page 170
14.8 Worked Problems......Page 172
References......Page 175
15.1 The Maximum Likelihood Method for Multiparameter Problems......Page 176
15.2 Regression Analysis with Non-constant Variance......Page 179
15.3 The Gibbs Phenomenon......Page 183
15.4 The Regularization Method......Page 185
15.5 Other Regularization Schemes......Page 187
15.6 Fitting Data with Errors in Both x and y......Page 188
15.7 Non-linear Parameters......Page 190
15.8 Optimizing a Data Set with Signal and Background......Page 194
15.10 Worked Problems......Page 197
15.11 Exercises......Page 200
References......Page 202
16.1 Handling Nuisance Variables......Page 203
16.2 Constraints on Nuisance Variables......Page 204
16.4 Iterations and Correlation Matrices......Page 206
16.5 Putting Together Several Probability Estimates......Page 212
References......Page 215
17.1 The Jacknife......Page 216
17.2 Making the Distribution Function of the Estimate Close …......Page 217
17.3 Estimating in Advance the Number of Events Needed for an Experiment......Page 222
References......Page 225
18.2 Spline Functions......Page 226
18.3 B-Splines......Page 228
18.4 Unfolding Data......Page 229
18.5.1 Uncertainty in the Data Statistics......Page 232
18.5.3 Uncertainty in the Theory......Page 233
18.5.4 Another Suggested Unfolding Technique......Page 234
References......Page 235
19.2 Tests on the Distribution Function......Page 236
19.3 Tests Based on the Binomial Distribution......Page 242
19.4 Tests Based on the Distributions of Deviations In Individual Bins of a Histogram......Page 244
19.5 Exercises......Page 245
References......Page 246
20.1 Neural Networks......Page 247
20.1.1 Gradient Descent......Page 249
20.1.2 Back Propagation......Page 250
20.1.3 Some Neural Net Variations......Page 251
20.2 Deep Learning......Page 253
20.2.1 Generative Adversarial Networks......Page 255
20.3 Boosted Decision Trees......Page 256
References......Page 260
Appendix A Brief Notes on Getting Started with CERN Root......Page 262
Appendix B Obtaining Pseudorandom Numbers in C++......Page 270
Appendix C Some Useful Spline Functions......Page 271
Index......Page 278