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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