Joseph Blitzstein, Jessica Hwang, (Blitzstein and Hwang 2019)

## Thoughts¶

### Skeleton¶

#### 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
• 2.9 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