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
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
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
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
Joint, marginal, and conditional D LOTUS Covariance and
correlation Multinomial Multivariate Normal Recap R Exercises
Change of variables Convolutions Beta Gamma Beta-Gamma
connections Order statistics Recap R Exercises
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 property and transition matrix Classification of states
Stationary distribution Reversibility Recap R Exercises
Markov Chain Monte Carlo
Metropolis-Hastings Recap R Exercises
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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