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Data 140
Introduction
Course Home
To the Student
Chapters
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
1.6. Exercises
2. Calculating Chances
2.1. Addition
2.2. Examples
2.3. Multiplication
2.4. More Examples
2.5. Updating Probabilities
2.6. Exercises
3. Random Variables
3.1. Functions on an Outcome Space
3.2. Distributions
3.3. Equality
3.4. Exercises
4. Relations Between Variables
4.1. Joint Distributions
4.2. Examples
4.3. Marginal Distributions
4.4. Conditional Distributions
4.5. Dependence and Independence
4.6. Exercises
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
5.5. Exercises
6. Random Counts
6.1. The Binomial Distribution
6.2. Examples
6.3. Multinomial Distribution
6.4. The Hypergeometric, Revisited
6.5. Odds Ratios
6.6. The Law of Small Numbers
6.7. Exercises
7. Poissonization
7.1. Poisson Distribution
7.2. Poissonizing the Binomial
7.3. Poissonizing the Multinomial
7.4. Exercises
8. Expectation
8.1. Definition
8.2. Applying the Definition
8.3. Expectations of Functions
8.4. Additivity
8.5. Method of Indicators
8.6. Exercises
9. Conditioning, Revisited
9.1. Probability by Conditioning
9.2. Expectation by Conditioning
9.3. Expected Waiting Times
9.4. Exercises
10. Markov Chains
10.1. Transitions
10.2. Deconstructing Chains
10.3. Long Run Behavior
10.4. Examples
11. Markov Chain Monte Carlo
11.1. Balance and Detailed Balance
11.2. Code Breaking
11.3. Metropolis Algorithm
11.4. Exercises
12. Standard Deviation
12.1. Definition
12.2. Prediction and Estimation
12.3. Tail Bounds
12.4. Heavy Tails
12.5. Exercises
13. Variance Via Covariance
13.1. Covariance
13.2. Properties of Covariance
13.3. Sums of Independent Variables
13.4. Symmetry and Indicators
13.5. Finite Population Correction
13.6. Exercises
14. The Central Limit Theorem
14.1. Exact Distribution of a Sum
14.2. PGFs in NumPy
14.3. Central Limit Theorem
14.4. SciPy and Normal Curves
14.5. The Sample Mean
14.6. Confidence Intervals
14.7. Exercises
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
15.6. Exercises
16. Transformations
16.1. Linear Transformations
16.2. Monotone Functions
16.3. Simulation via the CDF
16.4. Two-to-One Functions
16.5. Exercises
17. Joint Densities
17.1. Probabilities and Expectations
17.2. Independence
17.3. Marginal and Conditional Densities
17.4. Beta Densities with Integer Parameters
17.5. Exercises
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
18.5. Exercises
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
19.5. Exercises
20. Approaches to Estimation
20.1. Maximum Likelihood
20.2. Independence, Revisited
20.3. Prior and Posterior
20.4. Exercises
21. The Beta and the Binomial
21.1. Updating and Prediction
21.2. The Beta-Binomial Distribution
21.3. Long Run Proportion of Heads
21.4. Exercises
22. Prediction
22.1. Conditional Expectation As a Projection
22.2. Least Squares Predictor
22.3. Variance by Conditioning
22.4. Examples
22.5. Exercises
23. Jointly Normal Random Variables
23.1. Random Vectors
23.2. Multivariate Normal Vectors
23.3. Multivariate Normal Density
23.4. Independence
23.5. Exercises
24. Simple Linear Regression
24.1. Least Squares Linear Predictor
24.2. Bivariate Normal Distribution
24.3. Regression and the Bivariate Normal
24.4. The Regression Equation
24.5. Exercises
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
Index
