Joseph Blitzstein, Jessica Hwang, (Blitzstein and Hwang 2019)
Summary
Thoughts
Notes
Skeleton
Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
1: Probability and counting
- 1.1 Why study probability?
- 1.2 Sample spaces and Pebble World
- 1.3 Naive definition of probability
- 1.4 How to count
- 1.5 Story proofs
- 1.6 Non-naive definition of probability
- 1.7 Recap
- 1.8 R
- 1.9 Exercises
2: Conditional probability
- 2.1 The importance of thinking conditionally
- 2.2 Definition and intuition
- 2.3 Bayes' rule and the law of total probability
- 2.4 Conditional probabilities are probabilities
- 2.5 Independence of events
- 2.6 Coherency of Bayes' rule
- 2.7 Conditioning as a problem-solving tool
- 2.8 Pitfalls and paradoxes
- 2.9 Recap
- 2.10 R
- 2.11 Exercises
3: Random variables and their distributions
- 3.1 Random variables
- 3.2 Distributions and probability mass functions
- 3.3 Bernoulli and Binomial
- 3.4 Hypergeometric
- 3.5 Discrete Uniform
- 3.6 Cumulative distribution functions
- 3.7 Functions of random variables
- 3.8 Independence of r.v.s
- 3.9 Connections between Binomial and Hypergeometric
- 3.10 Recap
- 3.11 R
- 3.12 Exercises
4: Expectation
- 4.1 Definition of expectation
- 4.2 Linearity of expectation
- 4.3 Geometric and Negative Binomial
- 4.4 Indicator r.v.s and the fundamental bridge
- 4.5 Law of the unconscious statistician (LOTUS)
- 4.6 Variance
- 4.7 Poisson
- 4.8 Connections between Poisson and Binomial
- 4.9 *Using probability and expectation to prove existence
- 4.10 Recap
- 4.11 R
- 4.12 Exercises
5: Continuous random variables
- 5.1 Probability density functions
- 5.2 Uniform
- 5.3 Universality of the Uniform
- 5.4 Normal
- 5.5 Exponential
- 5.6 Poisson processes
- 5.7 Symmetry of i.i.d. continuous r.v.s
- 5.8 Recap
- 5.9 R
- 5.10 Exercises
6: Moments
- 6.1 Summaries of a distribution
- 6.2 Interpreting moments
- 6.3 Sample moments
- 6.4 Moment generating functions
- 6.5 Generating moments with MGFs
- 6.6 Sums of independent r.v.s via MGFs
- 6.7 *Probability generating functions
- 6.8 Recap
- 6.9 R
- 6.10 Exercises
7: Joint distributions
- 7.1 Joint, marginal, and conditional
- 7.2 2D LOTUS
- 7.3 Covariance and correlation
- 7.4 Multinomial
- 7.5 Multivariate Normal
- 7.6 Recap
- 7.7 R
- 7.8 Exercises
8: Transformations
- 8.1 Change of variables
- 8.2 Convolutions
- 8.3 Beta
- 8.4 Gamma
- 8.5 Beta-Gamma connections
- 8.6 Order statistics
- 8.7 Recap
- 8.8 R
- 8.9 Exercises
9: Conditional expectation
- 9.1 Conditional expectation given an event
- 9.2 Conditional expectation given an r.v.
- 9.3 Properties of conditional expectation
- 9.4 *Geometric interpretation of conditional expectation
- 9.5 Conditional variance
- 9.6 Adam and Eve examples
- 9.7 Recap
- 9.8 R
- 9.9 Exercises
10: Inequalities and limit theorems
- 10.1 Inequalities
- 10.2 Law of large numbers
- 10.3 Central limit theorem
- 10.4 Chi-Square and Student-t
- 10.5 Recap
- 10.6 R
- 10.7 Exercises
11: Markov chains
- 11.1 Markov property and transition matrix
- 11.2 Classification of states
- 11.3 Stationary distribution
- 11.4 Reversibility
- 11.5 Recap
- 11.6 R
- 11.7 Exercises
12: Markov chain Monte Carlo
- 12.1 Metropolis-Hastings
- 12.2 Gibbs sampling
- 12.3 Recap
- 12.4 R
- 12.5 Exercises
13: Poisson processes
- 13.1 Poisson processes in one dimension
- 13.2 Conditioning, superposition, and thinning
- 13.3 Poisson processes in multiple dimensions
- 13.4 Recap
- 13.5 R
- 13.6 Exercises
A: Math
- A.1 Sets
- A.2 Functions
- A.3 Matrices
- A.4 Difference equations
- A.5 Differential equations
- A.6 Partial derivatives
- A.7 Multiple integrals
- A.8 Sums
- A.9 Pattern recognition
- A.10 Common sense and checking answers
B: R
- B.1 Vectors
- B.2 Matrices
- B.3 Math
- B.4 Sampling and simulation
- B.5 Plotting
- B.6 Programming
- B.7 Summary statistics
- B.8 Distributions