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# Material for Final Exam

## General Concepts and Methods

### Probability

• Chapter 1: Spaces, events, basic counting, exponential approximation
• Chapter 2: Addition and multiplication rules; conditioning and updating
• Chapter 5: Unions and intersections of several events
• Section 9.1: Probabilities by conditioning and recursion (discrete)
• Section 20.2: Probabilities by conditioning on a continuous variable
• Section 4.5, 20.2: Independence

### Distribution

• Chapter 3: Intro; equality versus equality in distribution
• Chapter 4, Lab 1: Joint, marginals, conditionals, independence (discrete case), total variation distance
• Section 5.3, 5.4: Random permutations and symmetry
• Section 15.1, 15.2, Lab 4: Density
• Section 6.1, 15.1, 16.3, Lab 4: CDF and inverse CDF
• Chapter 16, Lab 4, Section 23.3, Lab 6: Density of a transformation
• Chapter 17: Joint, marginal, and conditional densities; independence
• Chapter 14, Chapter 19: Distribution of sum
• Chapter 14, Section 15.3, 19.3: Central Limit Theorem

### Expectation

• Chapter 8: The crucial properties (discrete case) including method of indicators, expectations of functions, tail sum formula (see also geometric distribution)
• Section 12.3, 19.4: Tail bounds: Markov, Chebyshev, Chernoff
• Section 9.2, 9.3: Expectation by conditioning
• Section 15.3, 17.1, 20.2: Expectation using densities and joint densities, and by conditioning on a continuous variable
• Section 19.2: Moment generating function

### Variance

• Chapter 12: Definition and basic properties; linear transformations
• Chapter 13: Covariance; variance of a sum
• Section 24.2, Homework 7: Correlation and its properties
• Section 22.3, 22.4, Homework 10: Variance by conditioning, mixtures
• Section 23.1: Mean vector and covariance matrix of random vectors

### Estimation and Prediction

• Section 8.4: Unbiased estimators
• Section 14.5, 14.6: IID sample mean; confidence interval for population mean
• Homework 8, Homework 13: Unbiased estimator of a population variance; independence of normal sample mean and sample variance
• Section 20.1: Maximum likelihood estimate
• Section 20.3: Posterior density, MAP estimate
• Section 12.2, 22.1, 22.2: Expectation and conditional expectation as least squares predictors
• Section 24.1, 25.4: Least squares linear predictor

## Special Distributions

### Uniform $(a, b)$

• Section 15.3: Density, expectation, variance, CDF
• Section 16.3, Lab 4: Use in simulation
• Section 19.1: Density of sum

### Normal

• Section 14.3, 14.4: CLT; Normal cdf and inverse cdf
• Sections 14.6: Normal confidence intervals
• Section 16.1: Normal densities
• Section 18.1, 18.2, 18.4: Independent normal variables, linear combinations, squares, Rayleigh, chi-squared
• Section 19.3: Normal MGF, sums, CLT
• Section 24.2, 24.3, Lab 6: Bivariate normal, linear combinations, independence, regression
• Chapter 23, Section 25.4: Multivariate normal, linear combinations, independence, regression

### Gamma

• Section 15.4, 16.1: Exponential and scaling
• Homework 9: Gamma function, gamma density, mean, variance
• Section 18.3, 18.4: Gamma and chi-squared
• Section 19.2: Sums of independent gammas with the same rate
• Lab 5: Waiting times of arrivals in a Poisson process

## Omitted from the Final

• Chapters 10, 11
• Section 12.4
• Section 21.3
• Sections 25.1, 25.2, 25.3