Introduction to Probability, Second Edition

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**Probability and Counting**- 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
- Conditional Probability
- 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
- Random Variables and Their Distributions
- 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
- Expectation
- 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
- Continuous Random Variables
- Probability density functions Uniform Universality of the Uniform Normal Exponential Poisson processes Symmetry of iid continuous rvs Recap R Exercises
- Moments
- 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
- Joint Distributions
- Joint, marginal, and conditional D LOTUS Covariance and correlation Multinomial Multivariate Normal Recap R Exercises
- Transformations
- Change of variables Convolutions Beta Gamma Beta-Gamma connections Order statistics Recap R Exercises
- Conditional Expectation
- 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
- Inequalities and Limit Theorems
- Inequalities Law of large numbers Central limit theorem Chi-Square and Student-t Recap R Exercises
- Markov Chains
- Markov property and transition matrix Classification of states Stationary distribution Reversibility Recap R Exercises
- Markov Chain Monte Carlo
- Metropolis-Hastings Recap R Exercises
- Poisson Processes

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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