Weighing the odds : a course in probability and statistics

Front Cover......Page 1
Back Cover......Page 2
Title Page......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
1.1 Conditioning: an intuitive first look......Page 21
1.2 Sharpening our intuition......Page 38
1.3 Probability as Pure Maths......Page 43
1.4 Probability as Applied Maths......Page 45
1.5 First remarks on Statistics......Page 46
1.6 Use of Computers......Page 51
2 EVENTS AND PROBABILITIES......Page 55
2.1 Possible outcome omega, actual outcome omega^act; and Events......Page 56
2.2 Probabilities......Page 59
2.3 Probability and Measure......Page 62
3.1 Random Variables......Page 67
3.2 DFs, pmfs and pdfs......Page 70
3.3 Means in the case when Omega is finite......Page 76
3.4 Means in general......Page 79
3.5 Variances and Covariances......Page 85
4.1 Conditional probabilities......Page 93
4.2 Independence......Page 116
4.3 Laws of large numbers......Page 123
4.4 Random Walks: a first look......Page 136
4.5 A simple \'strong Markov principle\'......Page 147
4.6 Simulation of IID sequences......Page 150
5 GENERATING FUNCTIONS;  AND THE CENTRAL LIMIT THEOREM......Page 161
5.1 General comments on the
use of Generating Functions (GFs)......Page 162
5.2 Probability generating functions (pgfs)......Page 163
5.3 Moment Generating Functions (MGFs)......Page 166
5.4 The Central Limit Theorem (CLT)......Page 176
5.5 Characteristic Functions (CFs)......Page 186
6.1 Introduction......Page 189
6.2 Some commonsense Frequentist CIs......Page 193
6.3 Likelihood; sufficiency; exponential family......Page 201
6.4 Brief notes on Point Estimation......Page 207
6.5 Maximum-Likelihood Estimators (MLEs) and
associated CIs......Page 212
6.6 Bayesian Confidence Intervals......Page 219
6.7 Hypothesis Testing — if you must......Page 242
7.1 Joint and conditional pmfs......Page 260
7.2 Jacobians......Page 263
7.3 Joint pdfs; transformations......Page 266
7.4 Conditional pdfs......Page 278
7.5 Multi-parameter Bayesian Statistics......Page 285
8.1 Overview and motivation......Page 303
8.2 The Orthonormality Principle and the F-test......Page 315
8.3 Five basic models: the Mathematics......Page 324
8.4 Goodness of fit; robustness; hierarchical models......Page 359
8.5 Multivariate Normal (MVN) Distributions......Page 385
9 SOME FURTHER PROBABILITY......Page 403
9.1 Conditional Expectation......Page 405
9.2 Martingales......Page 426
9.3 Poisson Processes (PPs)......Page 446
10 QUANTUM PROBABILITY  and QUANTUM COMPUTING......Page 460
10.1 Quantum Computing: a first look......Page 461
10.2 Foundations of Quantum Probability......Page 468
10.3 Quantum computing: a closer look......Page 483
10.4 Spin and Entanglement......Page 492
10.5 Spin and the Dirac equation......Page 505
10.6 Epilogue......Page 511
APPENDIX A: SOME PREREQUISITES  and ADDENDA......Page 515
APPENDIX B: DISCUSSION OF SOME SELECTED EXERCISES......Page 522
APPENDIX C: TABLES......Page 534
APPENDIX D: A SMALL SAMPLE OF THE LITERATURE......Page 539
Bibliography......Page 545
Index......Page 559