Fundamental Probability

Preface. A note to the student (and instructor). A note to the instructor (and student). Acknowledgements. Introduction. PART I: BASIC PROBABILITY. 1. Combinatorics. 1.1 Basic counting. 1.2 Generalized binomial coefficients. 1.3 Combinatoric identities and the use of induction. 1.4 The binomial and multinomial theorems. 1.4.1 The binomial theorem. 1.4.2 An extension of the binomial theorem. 1.4.3 The multinomial theorem. 1.5 The gamma and beta functions. 1.5.1 The gamma function. 1.5.2 The beta function. 1.6 Problems. 2. Probability spaces and counting. 2.1 Introducing counting and occupancy problems. 2.2 Probability spaces. 2.2.1 Introduction. 2.2.2 Definitions. 2.3 Properties. 2.3.1 Basic properties. 2.3.2 Advanced properties. 2.3.3 A theoretical property. 2.4 Problems. 3. Symmetric spaces and conditioning. 3.1 Applications with symmetric probability spaces. 3.2 Conditional probability and independence. 3.2.1 Total probability and Bayes' rule. 3.2.2 Extending the law of total probability. 3.2.3 Statistical paradoxes and fallacies. 3.3 The problem of the points. 3.3.1 Three solutions. 3.3.2 Further gambling problems. 3.3.3 Some historical references. 3.4 Problems. PART II: DISCRETE RANDOM VARIABLES. 4. Univariate random variables. 4.1 Definitions and properties. 4.1.1 Basic definitions and properties. 4.1.2 Further definitions and properties. 4.2 Discrete sampling schemes. 4.2.1 Bernoulli and binomial. 4.2.2 Hypergeometric. 4.2.3 Geometric and negative binomial. 4.2.4 Inverse hypergeometric. 4.2.5 Poisson approximations. 4.2.6 Occupancy distributions. 4.3 Transformations. 4.4 Moments. 4.4.1 Expected value of X. 4.4.2 Higher-order moments. 4.4.3 Jensen's inequality. 4.5 Poisson processes. 4.6 Problems. 5. Multivariate random variables. 5.1 Multivariate density and distribution. 5.1.1 Joint cumulative distribution functions. 5.1.2 Joint probability mass and density functions. 5.2 Fundamental properties of multivariate random variables. 5.2.1 Marginal distributions. 5.2.2 Independence. 5.2.3 Exchangeability. 5.2.4 Transformations. 5.2.5 Moments. 5.3 Discrete sampling schemes. 5.3.1 Multinomial. 5.3.2 Multivariate hypergeometric. 5.3.3 Multivariate negative binomial. 5.3.4 Multivariate inverse hypergeometric. 5.4 Problems. 6. Sums of random variables. 6.1 Mean and variance. 6.2 Use of exchangeable Bernoulli random variables. 6.2.1 Examples with birthdays. 6.3 Runs distributions. 6.4 Random variable decomposition. 6.4.1 Binomial, negative binomial and Poisson. 6.4.2 Hypergeometric. 6.4.3 Inverse hypergeometric. 6.5 General linear combination of two random variables. 6.6 Problems. PART III: CONTINUOUS RANDOM VARIABLES. 7. Continuous univariate random variables. 7.1 Most prominent distributions. 7.2 Other popular distributions. 7.3 Univariate transformations. 7.3.1 Examples of one-to-one transformations. 7.3.2 Many-to-one transformations. 7.4 The probability integral transform. 7.4.1 Simulation. 7.4.2 Kernel density estimation. 7.5 Problems. 8. Joint and conditional random variables. 8.1 Review of basic concepts. 8.2 Conditional distributions. 8.2.1 Discrete case. 8.2.2 Continuous case. 8.2.3 Conditional moments. 8.2.4 Expected shortfall. 8.2.5 Independence. 8.2.6 Computing probabilities via conditioning. 8.3 Problems. 9. Multivariate transformations. 9.1 Basic transformation. 9.2 The t and F distributions. 9.3 Further aspects and important transformations. 9.4 Problems. Appendix A. Calculus review. A.0 Recommended reading. A.1 Sets, functions and fundamental inequalities. A.2 Univariate calculus. A.2.1 Limits and continuity. A.2.2 Differentiation. A.2.3 Integration. A.2.4 Series. A.3 Multivariate calculus. A.3.1 Neighborhoods and open sets. A.3.2 Sequences, limits and continuity. A.3.3 Differentiation. A.3.4 Integration. Appendix B. Notation tables. Appendix C. Distribution tables. References. Index.