2nd Edition

Introduction to Probability, Second Edition

By Joseph K. Blitzstein, Jessica Hwang Copyright 2019
    634 Pages
    by Chapman & Hall

    Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. 

    The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces.

    The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.

    The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources. 

    Supplementary material is available on Joseph Blitzstein’s website www. stat110.net. The supplements include:
    Solutions to selected exercises
    Additional practice problems
    Handouts including review material and sample exams Animations and interactive visualizations created in connection with the edX online version of Stat 110.
    Links to lecture videos available on ITunes U and YouTube There is also a complete instructor's solutions manual available to instructors who require the book for a course.

    1. Probability and Counting
    2. Why study probability?

      Sample spaces and Pebble World

      Naive definition of probability

      How to count

      Story proofs

      Non-naive definition of probability

      Recap

      R

      Exercises

    3. Conditional Probability
    4. The importance of thinking conditionally

      Definition and intuition

      Bayes' rule and the law of total probability

      Conditional probabilities are probabilities

      Independence of events

      Coherency of Bayes' rule

      Conditioning as a problem-solving tool

      Pitfalls and paradoxes

      Recap

      R

      Exercises

    5. Random Variables and Their Distributions
    6. Random variables

      Distributions and probability mass functions

      Bernoulli and Binomial

      Hypergeometric

      Discrete Uniform

      Cumulative distribution functions

      Functions of random variables

      Independence of rvs

      Connections between Binomial and Hypergeometric

      Recap

      R

      Exercises

    7. Expectation
    8. Definition of expectation

      Linearity of expectation

      Geometric and Negative Binomial

      Indicator rvs and the fundamental bridge

      Law of the unconscious statistician (LOTUS)

      Variance

      Poisson

      Connections between Poisson and Binomial

      *Using probability and expectation to prove existence

      Recap

      R

      Exercises

    9. Continuous Random Variables
    10. Probability density functions

      Uniform

      Universality of the Uniform

      Normal

      Exponential

      Poisson processes

      Symmetry of iid continuous rvs

      Recap

      R

      Exercises

    11. Moments
    12. Summaries of a distribution

      Interpreting moments

      Sample moments

      Moment generating functions

      Generating moments with MGFs

      Sums of independent rvs via MGFs

      *Probability generating functions

      Recap

      R

      Exercises

    13. Joint Distributions
    14. Joint, marginal, and conditional

      D LOTUS

      Covariance and correlation

      Multinomial

      Multivariate Normal

      Recap

      R

      Exercises

    15. Transformations
    16. Change of variables

      Convolutions

      Beta

      Gamma

      Beta-Gamma connections

      Order statistics

      Recap

      R

      Exercises

    17. Conditional Expectation
    18. Conditional expectation given an event

      Conditional expectation given an rv

      Properties of conditional expectation

      *Geometric interpretation of conditional expectation

      Conditional variance

      Adam and Eve examples

      Recap

      R

      Exercises

    19. Inequalities and Limit Theorems
    20. Inequalities

      Law of large numbers

      Central limit theorem

      Chi-Square and Student-t

      Recap

      R

      Exercises

    21. Markov Chains
    22. Markov property and transition matrix

      Classification of states

      Stationary distribution

      Reversibility

      Recap

      R

      Exercises

    23. Markov Chain Monte Carlo
    24. Metropolis-Hastings

      Recap

      R

      Exercises

    25. Poisson Processes

    Poisson processes in one dimension

    Conditioning, superposition, thinning

    Poisson processes in multiple dimensions

    Recap

    R

    Exercises

    A Math

    A Sets

    A Functions

    A Matrices

    A Difference equations

    A Differential equations

    A Partial derivatives

    A Multiple integrals

    A Sums

    A Pattern recognition

    A Common sense and checking answers

    B R

    B Vectors

    B Matrices

    B Math

    B Sampling and simulation

    B Plotting

    B Programming

    B Summary statistics

    B Distributions

    C Table of distributions

    Bibliography

    Index

    Biography

    Joseph K. Blitzstein, PhD, professor of the practice in statistics, Department of Statistics, Harvard University, Cambridge, Massachusetts, USA

    Jessica Hwang is a graduate student in the Stanford statistics department.