Probability

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 Linear regression 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.