Whirlwind

This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.-- Provided by Publisher Contents Preface To the instructor From gambling to an essential ingredient of modern science and society Chapter 1 Experiments with random outcomes 1.1 Sample spaces and probabilities 1.2 Random sampling 1.3 Infinitely many outcomes 1.4 Consequences of the rules of probability 1.5 Random variables: a first look 1.6 Finer points ♣ Exercises Chapter 2 Conditional probability and independence 2.1 Conditional probability 2.2 Bayes’ formula 2.3 Independence 2.4 Independent trials 2.5 Further topics on sampling and independence 2.6 Finer points ♣ Exercises Chapter 3 Random variables 3.1 Probability distributions of random variables 3.2 Cumulative distribution function 3.3 Expectation 3.4 Variance 3.5 Gaussian distribution 3.6 Finer points ♣ Exercises Chapter 4 Approximations of the binomial distribution 4.1 Normal approximation 4.2 Law of large numbers 4.3 Applications of the normal approximation 4.4 Poisson approximation 4.5 Exponential distribution 4.6 Poisson process ± 4.7 Finer points ♣ Exercises Chapter 5 Transforms and transformations 5.1 Moment generating function 5.2 Distribution of a function of a random variable 5.3 Finer points ♣ Exercises Chapter 6 Joint distribution of random variables 6.1 Joint distribution of discrete random variables 6.2 Jointly continuous random variables 6.3 Joint distributions and independence 6.4 Further multivariate topics ± 6.5 Finer points ♣ Exercises Chapter 7 Sums and symmetry 7.1 Sums of independent random variables 7.2 Exchangeable random variables 7.3 Poisson process revisited ± Exercises Chapter 8 Expectation and variance in the multivariate setting 8.1 Linearity of expectation 8.2 Expectation and independence 8.3 Sums and moment generating functions 8.4 Covariance and correlation 8.5 The bivariate normal distribution ± 8.6 Finer points ♣ Exercises Chapter 9 Tail bounds and limit theorems 9.1 Estimating tail probabilities 9.2 Law of large numbers 9.3 Central limit theorem 9.4 Monte Carlo method ± 9.5 Finer points ♣ Exercises Chapter 10 Conditional distribution 10.1 Conditional distribution of a discrete random variable 10.2 Conditional distribution for jointly continuous random variables 10.3 Conditional expectation 10.4 Further conditioning topics ± 10.5 Finer points ♣ Exercises Appendix A Things to know from calculus Appendix B Set notation and operations Appendix C Counting Appendix D Sums, products and series Appendix E Table of values for ±(x) Appendix F Table of common probability distributions Answers to selected exercises Bibliography Index This textbook is an introduction to probability theory with the right balance between mathematical precision, probabilistic intuition, and concrete applications. It is intended for students with a calculus background, as well as for independent learners and readers. The material is covered precisely, while avoiding excessive technical details.