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Final Exam Contents

Prob 140 Spring 2018

A. Adhikari

Material for Final Exam

General Concepts and Methods

Probability

Chapter 1, Lab 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
Sections 4.5, 20.3: Independence

Distribution

Chapter 3: Intro; equality versus equality in distribution
Chapter 4: Joint, marginals, conditionals, independence (discrete case)
Sections 5.3, 5.4: Random permutations and symmetry
Sections 15.1, 15.2: Density
Section 15.1 and Lab 9: CDF and inverse CDF
Chapter 16: Density of a transformation
Chapter 17: Joint, marginal, and conditional densities; independence
Chapters 14, 19: Distribution of sum
Section 14.3, 14.4, 15.3, 19.3: Central Limit Theorem

Expectation

Chapter 8: The crucial properties (discrete case) including method of indicators and expectations of functions
Lab 4 Parts 1-3: Tail sum formula and applications; see also geometric distribution
Section 12.3, 19.4: Bounds: Markov, Chebyshev, Chernoff
Section 9.2, 9.3: Expectation by conditioning
Section 15.3, 17.1: Expectation using densities and joint densities
Section 14.1, 14.2, 21.2: Generating functions

Variance

Chapter 12: Intro, linear transformations
Chapter 13: Covariance; variance of a sum
Lab 8 Part 4: Application of mean and variance of simple random sample sum
Lab 11: Application of method of indicators
Homework 12: Correlation and its properties
Sections 22.2, 22.3: Variance by conditioning, mixtures
Sections 23.1, 25.1: Mean and covariance for random vectors

Estimation and Prediction

Section 8.2: Unbiased estimates
Sections 14.4, 14.5: IID sample mean; confidence interval for population mean
Section 20.1: Maximum likelihood estimate
Section 20.2: Posterior density, MAP estimate
Sections 12.2, 22.1, 22.4: Expectation and conditional expectation as least squares predictors
Sections 24.2, 25.2: Least squares linear predictor

Special Distributions

Random Counts

Sections 8.1, 12.1: Uniform on 1, 2, …, n
Sections 6.1, 6.2, Chapter 7, 13.2, 14.3, Chapter 21: Bernoulli, binomial and multinomial
Sections 6.3, 8.2, 13.3, 13.4, Lab 2: Hypergeometric
Section 6.4, 6.5, Chapter 7, Sections 8.1, 8.3, 12.1, 19.2 and related homework: Poisson
Homework 4, Sections 9.3, 22.3: Geometric

Uniform $(a, b)$

Section 15.3, 19.1: Density, expectation, variance, CDF, density of sum

Beta

Section 17.4: Integer parameters; uniform order statistics
Chapter 21: Relation with binomial; beta-binomial distribution

Normal

Section 14.3: CLT
Sections 14.4, 14.5: Normal confidence intervals
Section 16.1: Normal densities
Sections 18.1, 18.2, 18.4: Independent normal variables, linear combinations, squares, Rayleigh, chi squared
Section 19.3: Normal MGF, sums, CLT
Chapter 24, Lab 10: Bivariate normal, regression, independence, linear combinations
Chapters 23, 25: Multivariate normal, regression

Gamma

Section 15.4: Exponential
Homework 10: Gamma function, gamma density, mean, variance
Sections 18.3, 18.4: Gamma and chi squared
Sections 19.2: Sums of independent gammas with the same rate

Omitted from Final

Chapters 10, 11
Section 12.4