Joseph K. Blitzstein, Jessica Hwang | Introduction to Probability

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

C: Table of distributions

References

Index

Bibliography