Introduction to Probability 2nd Edition by Joseph Blitzstein,Jessica Hwang 9781466575578 1466575573

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