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Introduction to Probability, Second Edition


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Table of Contents

  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

About the Author

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.

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