Preface ..................................................... ix
1 Introduction ................................................. 1
1.1 Measuring g, the coefficient of acceleration due to
gravity ................................................. 1
1.2 Verification of Ohm's law ............................... 5
1.3 Measuring the half-life of an isotope ................... 7
1.4 Summary ................................................ 10
2 Sets ........................................................ 12
2.1 Relationships between sets ............................. 13
2.2 Summary ................................................ 17
Exercises ................................................... 18
3 Probability ................................................. 20
3.1 Elementary rales ....................................... 21
3.2 Bayesian probability ................................... 21
3.3 Classic approach ....................................... 24
3.4 Frequentist probability ................................ 25
3.5 Probability density functions .......................... 26
3.6 Likelihood ............................................. 27
3.7 Case studies ........................................... 27
3.8 Summary ................................................ 32
Exercises ................................................... 33
4 Visualising and quantifying the properties of data .......... 35
4.1 Visual representation of data .......................... 35
4.2 Mode, median, mean ..................................... 37
4.3 Quantifying the spread of data ......................... 39
4.4 Presenting a measurement ............................... 41
4.5 Skew ................................................... 43
4.6 Measurements of more than one observable ............... 44
4.7 Case study ............................................. 52
4.8 Summary ................................................ 53
Exercises ................................................... 53
5 Useful distributions ........................................ 56
5.1 Expectation values of probability density functions .... 57
5.2 Binomial distribution .................................. 57
5.3 Poisson distribution ................................... 62
5.4 Gaussian distribution .................................. 65
5.5 x2 distribution ........................................ 67
5.6 Computational issues ................................... 68
5.7 Summary ................................................ 70
Exercises ................................................... 70
6 Uncertainty and errors ...................................... 72
6.1 The nature of errors ................................... 72
6.2 Combination of errors .................................. 75
6.3 Binomial error ......................................... 79
6.4 Averaging results ...................................... 81
6.5 Systematic errors and systematic bias .................. 82
6.6 Blind analysis technique ............................... 84
6.7 Case studies ........................................... 85
6.8 Summary ................................................ 90
Exercises ................................................... 91
7 Confidence intervals ........................................ 93
7.1 Two-sided intervals .................................... 93
7.2 Upper and lower limit calculations ..................... 94
7.3 Limits for a Gaussian distribution ..................... 96
7.4 Limits for a Poisson distribution ...................... 98
7.5 Limits for a binomial distribution .................... 100
7.6 Unified approach to analysis of small signals ......... 101
7.7 Monte Carlo method .................................... 105
7.8 Case studies .......................................... 106
7.9 Summary ............................................... 111
Exercises .................................................. 112
8 Hypothesis testing ......................................... 114
8.1 Formulating a hypothesis .............................. 114
8.2 Testing if the hypothesis agrees with data ............ 115
8.3 Testing if the hypothesis disagrees with data ......... 117
8.4 Hypothesis comparison ................................. 117
8.5 Testing the compatibility of results .................. 119
8.6 Establishing evidence for, or observing a new effect .. 120
8.7 Case studies .......................................... 124
8.8 Summary ............................................... 125
Exercises .................................................. 126
9 Fitting .................................................... 128
9.1 Optimisation .......................................... 128
9.2 The least squares or χ2 fit ........................... 131
9.3 Linear least-squares fit .............................. 134
9.4 Maximum-likelihood fit ................................ 136
9.5 Combination of results ................................ 140
9.6 Template fitting ...................................... 142
9.7 Case studies .......................................... 142
9.8 Summary ............................................... 150
Exercises .................................................. 151
10 Multivariate analysis ...................................... 153
10.1 Cutting on variables .................................. 154
10.2 Bayesian classifier ................................... 157
10.3 Fisher discriminant ................................... 158
10.4 Artificial neural networks ............................ 162
10.5 Decision trees ........................................ 169
10.6 Choosing an MVA technique ............................. 171
10.7 Case studies .......................................... 174
10.8 Summary ............................................... 177
Exercises .................................................. 178
Appendix A Glossary ....................................... 181
Appendix В Probability density functions .................. 186
Appendix С Numerical integration methods .................. 198
Appendix D Solutions ...................................... 201
Appendix E Reference tables ............................... 207
References ................................................. 216
Index ...................................................... 218
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