Clark University
                                        Math 217/Econ 360, Probability and Statistics
Fall 2014
Prof. D. Joyce, BP 322, 793-7421
Department of Mathematics and Computer Science
Clark University

Pierre-Simon Laplace (1749–1827)
Portraint of Pierre Simon Laplace
click for source Portrait of Laplace at the Palace of Versailles
Date 1842, posthumous portrait by Sophie Feytaud

[This course page is obsolete.  I'll update it next time I teach the course.]

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Course information

Class notes, quizzes, tests, homework assignments

The dates for the discussion topics and the assignments are tentative. They will change as the course progresses.

  1. Monday, Aug 25. Welcome and introduction to the class
    Intro to probability via discrete uniform probabilities. Symmetry. Frequency.
    Simulations and random walks

  2. Wednesday, Aug 27.
    Background on sets. Unions, intersections, complements, distributivity, DeMorgan's laws. Product, power sets. Countable and uncountable infinities.
    Combinatorics. Principle of inclusion and exclusion, multiplicative principle, permutations, factorials and Sterling's approximation
    Permutation generating applet
    Assignment 1 due next Wednesday.

  3. Friday, Aug 29. Binomial coefficients.
    Combinations, Pascal's triangle, multinomial coefficients, stars & bars, combinatorial proofs
    The BinomialPlot and GaltonBoard applets

    Monday, Sep 1. Labor Day. No classes

  4. Wednesday, Sep 3. Axioms for probability distributions.
    Probability mass mass functions for discrete distributions, density functions for continuous distributions. Cumulative distribution functions. Sample spaces, axioms, and properties.
    Assignment 1 due.
    Assignment 2 due next Wednesday.

  5. Friday, Sep 5. Uniform finite probabilities
    Odds. Repeated trials. Sampling with replacement. The birthday problem.
    The Birthday applet

  6. Monday, Sep 8. Proofs of properties of probability distributions from the axioms.
    Conditional probability. Xox.

  7. Wednesday, Sep 10.
    More on conditional probability: definition of conditional probability, the multiplication rule
    Assignment 2 due.
    Assignment 3 due next Wednesday.

  8. Friday, Sep 12. Bayes' formula. Examples, tree diagrams

  9. Monday, Sep 15.
    Independent events. Definition, product spaces, independence of more than two events, joint random variables, random samples, i.i.d. random variables
    Bertrand's box paradox

  10. Wednesday, Sep 17.
    The Bernoulli process. Sampling with replacement. Binomial distribution, geometric distribution, negative binomial distribution, hypergeometric distribution. Sampling without replacement
    Assignment 3 due.
    Assignment 4 due next Wednesday.

  11. Friday, Sep 19.
    Discrete random variables. Probability mass functions, cumulative distribution functions. Various graphs and charts
    Expectation for discrete random variables. Definition, expectation for the binomial and geometric distributions. St. Petersburg paradox.

  12. Monday, Sep 22.
    Select the date for the first test. It will cover chapter 1 through most of chapter 4.
    More on expectation. Properties of expectation.
    Variance for discrete random variables. Definition and properties.

  13. Wednesday, Sep 24.
    Variance of the binomial and geometric distributions.
    Assignment 4 due.
    Assignment 5 due next Wednesday.

  14. Friday, Sep 26.
    Continuous probability. Monte Carlo estimates. Introduction to the Poisson process and the normal distribution. Statement of the central limit theorem.

  15. Monday, Sep 29.
    Density functions. Density as the derivative of the c.d.f., and the c.d.f. as the integral of density.

  16. Wednesday, Oct 1.
    Assignment 5 due.

  17. Friday, Oct 3.
    Examples of continuous distributions, functions of random variables, the Cauchy distribution.

  18. Monday, Oct 6. Review for first test
    Sample first test

  19. Wednesday, Oct 8.
    First test. Covers chapter 1 through 4. Answers. Extra answers.

  20. Friday, Oct 10.
    The Poisson process. The Poisson, exponential, gamma, and beta distributions. Axioms for the Poisson process.

    Monday, Oct 13. Fall break. No classes.

  21. Wednesday, Oct 15.
    Expectation and variance for continuous random variables. Definitions and properties. Expectation and variance for uniform and continuous distributions. Lack of expectation and variance for the Cauchy distribution.
    Assignment 6 due next Wednesday.

  22. Friday, Oct 17. The normal distribution, table for the c.d.f. of the standard normal distribution, the normal approximation to the binomial distribution. DeMoivre's 1733 proof for the first instance of the Central Limit Theorem.

  23. Monday, Oct 20.
    Joint distributions. Independent random variables. Joint c.d.f.'s and joint density functions.

  24. Wednesday, Oct 22. Partial derivatives and multiple integrals relating to joint distributions.
    Assignment 6 due.

  25. Friday, Oct 24. Sums and convolution. The discrete case.
    Assignment 7 due next Wednesday.

  26. Monday, Oct 27. More on convolution. The continuous case. Gamma distributions as convolution of exponential distributions. Normal distributions convolute to other normal distributions.

  27. Wednesday, Oct 29.
    Conditional distributions. Conditional cumulative distribution functions, conditional probability mass functions, conditional probability density functions
    Assignment 7 due.

  28. Friday, Oct 31.
    Covariance and correlation. Connection of covariance and variance, properties of covariance including bilinearity.
    See Spurious Correlations to see that correlations often occur without associated causation.

  29. Monday, Nov 3.
    Order statistics

  30. Wednesday, Nov 5. Review for second test
    Sample second test

  31. Friday, Nov 7. Second test. On 4.7, 5, and 6.1–6.5 (through conditional distributions). Answers. Extra answers.

  32. Monday, Nov 10.
    Moments and the moment generating function

  33. Wednesday, Nov 12. Properties and examples of the moment generating function.

  34. Friday, Nov 14. Joint probability distributions of multivariate functions, the Jacobian.
    Notes from multivariate calculus about the Jacobians
    Assignment 8 due next Wednesday.

  35. Monday, Nov 17.
    A proof of the central limit theorem

  36. Wednesday, Nov 19. Discussion of statistical inference.
    Assignment 8 due. Answers

  37. Friday, Nov 21.
    Bayesian statistics Part I: an example, the basic principle, the Bernoulli process

  38. Monday, Nov 24. Maximum likelihood estimators
    Old notes from 218: Discrete case and Continuous case

  39. Monday, Dec 1.
    Bayesian statistics Part II: Bayes pool table, conjugate priors for the Bernoulli process, point estimators, interval estimators

  40. Wednesday, Dec 3.
    Bayesian statistics Part III: the Poisson process & its conjugate priors
    Bayesian statistics Part IV: the normal distribution with known variance

  41. Friday, Dec 5.
    Bayesian statistics Part V: the normal distribution with unknown variance

  42. Monday, Dec 8. Review.
    Final will cover chapters 1–8 except sections 6.8, 7.6, 7.8, 7.9, 8.2, 8.4, 8.5. Also there will be an essay question on Bayesian statistics.
    Sample final exam

  43. Final exam.
    Dec 16. 10:30. JC 220. Alternate Dec 12. morning, meet at my office sometime between 9 and noon.
Other information.

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David E. Joyce