Probability

This book is an elementary and practical introduction to probability theory. It differs from other introductory texts in two important respects. First, the per­ sonal (or subjective) view of probability is adopted throughout. Second, emphasis is placed on how values are assigned to probabilities in practice, i.e. the measurement of probabilities. The personal approach to probability is in many ways more natural than other current formulations, and can also provide a broader view of the subject. It thus has a unifying effect. It has also assumed great importance recently because of the growth of Bayesian Statistics. Personal probability is essential for modern Bayesian methods, and it can be difficult for students who have learnt a different view of probability to adapt to Bayesian thinking. This book has been produced in response to that difficulty, to present a thorough introduction to probability from scratch, and entirely in the personal framework.

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1 Probability and its laws.- 1.1 Uncertainty and probability.- 1.2 Direct measurement.- Exercises 1(a).- 1.3 Betting behaviour.- 1.4 Fair bets.- 1.5 The Addition Law.- Exercises 1(b).- 1.6 The Multiplication Law.- 1.7 Independence.- Exercises 1(c).- 2 Probability measurements.- 2.1 True probabilities.- Exercises 2(a).- 2.2 Elaboration.- Exercises 2(b).- 2.3 The disjunction theorem.- Exercises 2(c).- 2.4 The sum theorem.- Exercises 2(d).- 2.5 Partitions.- 2.6 Symmetry probability.- Exercises 2(e).- 3 Bayes' theorem.- 3.1 Extending the argument.- Exercises 3(a).- 3.2 Bayes' theorem.- 3.3 Learning from experience.- Exercises 3(b).- 3.4 Zero probabilities in Bayes' theorem.- 3.5 Example: disputed authorship.- 4 Trials and deals.- 4.1 The product theorem.- 4.2 Mutual independence.- Exercises 4(a).- 4.3 Trials.- 4.4 Factorials and combinations.- Exercises 4(b).- 4.5 Binomial probabilities.- Exercises 4(c).- 4.6 Multinomial probabilities.- Exercises 4(d).- 4.7 Deals.- 4.8 Probabilities from information.- Exercises 4(e).- 4.9 Properties of deals.- 4.10 Hypergeometric probabilities.- Exercises 4(f).- 4.11 Deals from large collections.- Exercises 4(g).- 5 Random variables.- 5.1 Definitions.- 5.2 Two or more random variables.- Exercises 5(a).- 5.3 Elaborations with random variables.- 5.4 Example: capture-recapture.- 5.5 Example: job applications.- Exercises 5(b).- 5.6 Mean and standard deviation.- Exercises 5(c).- 5.7 Measuring distributions.- 5.8 Some standard distributions.- Exercises 5(d).- 6 Distribution theory.- 6.1 Deriving standard distributions.- 6.2 Combining distributions.- Exercises 6(a).- 6.3 Basic theory of expectations.- 6.4 Further expectation theory.- Exercises 6(b).- 6.5 Covariance and correlation.- Exercises 6(c).- 6.6 Conditional expectations.- 6.7 Linearregression functions.- Exercises 6(d).- 7 Continuous distributions.- 7.1 Continuous random variables.- 7.2 Distribution functions.- Exercises 7(a).- 7.3 Density functions.- 7.4 Transformations and expectations.- Exercises 7(b).- 7.5 Standard continuous distributions.- Exercises 7(c).- 7.6 Two continuous random variables.- 7.7 Example: heat transfer.- Exercises 7(d).- 7.8 Random variables of mixed type.- Exercises 7(e).- 7.9 Continuous distribution theory.- Exercises 7(f).- 8 Frequencies.- 8.1 Exchangeable propositions.- 8.2 The finite characterization.- Exercises 8(a).- 8.3 De Finetti's theorem.- 8.4 Updating.- Exercises 8(b).- 8.5 Beta prior distributions.- Exercises 8(c).- 8.6 Probability and frequency.- 8.7 Calibration.- 9 Statistical models.- 9.1 Parameters and models.- 9.2 Exchangeable random variables.- Exercises 9(a).- 9.3 Samples.- 9.4 Measuring mean and variance.- Exercises 9(b).- 9.5 Exchangeable parametric models.- 9.6 The normal location model.- Exercises 9(c).- 9.7 The Poisson model.- 9.8 Linear estimation.- Exercises 9(d).- 9.9 Postscript.- Appendix - Solutions to exercises.