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Chapter 1: Fundamentals
1.1 Outcome Space and Events
1.2 Equally Likely Outcomes
1.3 Collisions in Hashing
1.4 The Birthday Problem
1.5 An Exponential Approximation
Chapter 2: Calculating Chances
2.1 Addition
2.2 Examples
2.3 Multiplication
2.4 More Examples
2.5 Updating Probabilities
Chapter 3: Random Variables
3.1 Functions on an Outcome Space
3.2 Distributions
3.3 Equality
Chapter 4: Relations Between Variables
4.1 Joint Distributions
4.2 Marginal Distributions
4.3 Conditional Distributions
4.4 Updating Distributions
4.5 Dependence and Independence
Chapter 5: Collections of Events
5.1 Bounding the Chance of a Union
5.2 Inclusion-Exclusion
5.3 The Matching Problem
5.4 Sampling Without Replacement
Review Problem Set 1
Chapter 6: Random Counts
6.1 The Binomial Distribution
6.2 Examples
6.3 The Hypergeometric Distribution
6.4 Odds Ratios
6.5 The Law of Small Numbers
Chapter 7: Poissonization
7.1 Poissonizing the Binomial
7.2 Poissonizing the Multinomial
Chapter 8: Expectation
8.1 Definition
8.2 Additivity
8.3 Expectations of Functions
Review Problems: Set 2
Chapter 9: Conditioning, Revisited
9.1 Probability by Conditioning
9.2 Expectation by Conditioning
9.3 Expected Waiting Times
Chapter 10: Markov Chains
10.1 Transitions
10.2 Deconstructing Chains
10.3 Long Run Behavior
10.4 Examples
Chapter 11: Reversing Markov Chains
11.1 Detailed Balance
11.2 Reversibility
11.3 Code Breaking
11.4 Markov Chain Monte Carlo
Review Set on Conditioning and Markov Chains
Chapter 12: Standard Deviation
12.1 Definition
12.2 Prediction and Estimation
12.3 Tail Bounds
12.4 Heavy Tails
Chapter 13: Variance Via Covariance
13.1 Properties of Covariance
13.2 Sums of IID Samples
13.3 Sums of Simple Random Samples
13.4 Finite Population Correction
Chapter 14: The Central Limit Theorem
14.1 Exact Distribution
14.2 PGFs in NumPy
14.3 Central Limit Theorem
14.4 The Sample Mean
14.5 Confidence Intervals
Chapter 15: Continuous Distributions
15.1 Density and CDF
15.2 The Meaning of Density
15.3 Expectation
15.4 Exponential Distribution
15.5 Calculus in SymPy
Review Problems: Set 3
Chapter 16: Transformations
16.1 Linear Transformations
16.2 Monotone Functions
16.3 Two-to-One Functions
Chapter 17: Joint Densities
17.1 Probabilities and Expectations
17.2 Independence
17.3 Marginal and Conditional Densities
17.4 Beta Densities with Integer Parameters
Chapter 18: The Normal and Gamma Families
18.1 Standard Normal: The Basics
18.2 Sums of Independent Normal Variables
18.3 The Gamma Family
18.4 Chi-Squared Distributions
Review Problems: Set 4
Chapter 19: Distributions of Sums
19.1 The Convolution Formula
19.2 Moment Generating Functions
19.3 MGFs, the Normal, and the CLT
19.4 Chernoff Bound
Chapter 20: Approaches to Estimation
20.1 Maximum Likelihood
20.2 Prior and Posterior
20.3 Independence, Revisited
Chapter 21: The Beta and the Binomial
21.1 Updating and Prediction
21.2 The Beta-Binomial Distribution
21.3 Long Run Proportion of Heads
Chapter 22: Prediction
22.1 Conditional Expectation As a Projection
22.2 Variance by Conditioning
22.3 Examples
22.4 Least Squares Predictor
Chapter 23: Jointly Normal Random Variables
23.1 Random Vectors
23.2 Multivariate Normal Distribution
23.3 Linear Combinations
23.4 Independence
Chapter 24: Simple Linear Regression
24.1 Bivariate Normal Distribution
24.2 Least Squares Linear Predictor
24.3 Regression and the Bivariate Normal
24.4 The Regression Equation
Chapter 25: Multiple Regression
25.1 Bilinearity in Matrix Notation
25.2 Best Linear Predictor
25.3 Conditioning and the Multivariate Normal
25.4 Multiple Regression
Further Review Exercises
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