Welcome to the Stats 620 home page

Or, you can catch me in my office. My office is room 205C, Cockins Hall. My phone number is 292-5843. My office hours are on Monday, from 2:30 - 4:18. You can also catch me at other times.

The course TA is Kazuki Uematsu. Kazuki's office is room MA 454. His phone number is 292-9247. His office hour is Thursday, 2:00 - 3:00. His email is kazuki@stat.osu.edu.

Reading Assignments

• Week One.

  The first week of the course will cover material that is mostly review for you if you have had a course on theoretical probability and statistics. Watch out for the new bits--the notion of a Borel field and the move toward more formality in proofs and in solutions to problems.

• Lecture 1. Wednesday, September 23. Introduction.

  Introduction to the course, course policies. Definitions: experiment, sample space, outcome, event, intersection, union, complement, axioms of probability, sigma field/Borel field. Reading assignment: 1.1.

• Lecture 2. Friday, September 25. Probability Models.

  Probability as a function, the principle of insufficient reason, rules of probability, examples of counting problems, conditional probability. Reading assignment: 1.2 - 1.3.

• Week Two.

  The second week of the course will begin with conditional probability and will introduce Bayes' Theorem--the theorem which describes how we learn about the world. We will move from considering probabilities of events described in words to probabilities of events described by numbers--hence to random variables.

• Lecture 3. Monday, September 28. Conditional Probability.

  Probability examples, Bayes theorem, independence and dependence. Reading assignment: 1.3.

• Lecture 4. Wednesday, September 30. Random Variables - 1.

  Independence and dependence, discrete random variables, continuous random variables, cdf, pdf, pmf. Reading assignment: 1.4 - 1.5.

• Lecture 5. Friday, October 2. Random Variables - 2.

  More on random variables. Examples of discrete and continuous r.v.s. The geometric random variable. The uniform random variable. Families of random variables and parameters. Reading assignment: 1.6, 1.8.

• Week Three.

  The third week of the course will focus on random variables. We will cover basic facts about random variables, how we look at them (cdf and pdf), transformations, and moments.

• Lecture 6. Monday, October 5. Transformations.

  Transformations via the cdf method. Transformations via the pdf method. The probability integral transform. Non 1-1 transformations. Reading assignment: 2.1.

• Lecture 7. Wednesday, October 7. Expected Value.

  The mean. The expected value. Higher order moments. Rules for expectations. The binomial random variable. The variance. Reading assignment: 2.2.

• Lecture 8. Friday, October 9. Moment Generating Functions.

  The moment generating function. Uniqueness of moment generating functions. The moment generating function for a linear transformation. Convergence of moment generating functions. Reading assignment: 2.3. You may also wish to read 2.4, although reading this section is optional.

• Week Four.

  We will begin the fourth week with a quiz (well, not quite, as the quiz will be at the end of class). We will further explore the moment generating function and learn about connections with transformations of random variables. We will also begin a systematic description of families of random variables.

• Lecture 9. Monday, October 12. QUIZ I.

  A selection of discrete random variables. The binomial distribution. The Bernoulli distribution. The hypergeometric distribution. QUIZ I during the last 20 minutes of class. The quiz "covers" chapter 1 and section 2.1. Reading assignment: 3.1-3.2.

• Lecture 10. Wednesday, October 14. Discrete Distributions.

  The geometric distribution. The memoryless property. The Poisson distribution. Reading assignment: 3.2, 3.8.1.

• Lecture 11. Friday, October 16. Continuous Distributions 1.

  Recursion (dice, Poisson probabilities). The Poisson process. The exponential distribution. The gamma distribution. Reading assignment: 3.3.

• Week Five.

  The fifth week of the course will focus on further families of random variables. Most of the effort this week will be devoted to continuous random variables. We will also discuss the exponential family--a hyper-family into which most of our familiar families of random variables fall. An amazing variety of results hold for all random variables in the exponential family.

• Lecture 12. Monday, October 19. Continuous Distributions 2.

  The chi-squared distribution. The beta distribution. The normal distribution. The normal approximation to the binomial distribution. The continuity correction. Other approximations. Reading assignment: 3.3.

• Lecture 13. Wednesday, October 21. Exponential Families.

  The Cauchy distribution. Exponential families. Definition of exponential families. Examples of exponential families. Results for exponential families. Reading assignment: 3.4.

• Lecture 14. Friday, October 23. Inequalities.

  Location-scale families. Chebyshev's inequality. Random vectors. Joint probability mass function. Marginal probability mass function. Reading assignment: 3.5-3.6.

• Week Six.

  The sixth week of the course will take us through the midterm exam. We will begin our study of groups of random variables, covering basic definitions dependence/independence, and examining consequences of independence.

• Lecture 15. Monday, October 26. Joint distributions.

  Joint probability density function. Joint cumulative distribution function. Marginal probability density function. Conditional distributions. Independence. Reading assignment: 4.1-4.2.

• Lecture 16. Wednesday, October 28. Change of variables.

  More on independence. Sums of independent random variables. The bivariate change of variables. The Jacobian. Examples. Reading assignment: 4.3.

• Lecture 17. Friday, October 30. MIDTERM EXAM.

  Midterm Exam.

• Week Seven.

  The seventh week of the course will continue to look at collections of random variables. We'll look at changes of variables. We will see the hierarchical model which leads to the famous decomposition of means and variances through conditional expectation. We will also look at measures of dependence between pairs of random variables.

• Lecture 18. Monday, November 2. The hierarchical model.

  The hierarchical model. The binomial-binomial model. The beta-binomial model. Examples of mixture distributions. Reading assignment: 4.4.

• Lecture 19. Wednesday, November 4. Measures of dependence.

  Exam commentary. The covariance. Properties of the covariance. The correlation coefficient. Properties of correlation. Reading assignment: 4.5.

• Lecture 20. Friday, November 6. Multivariate random vectors.

  Sums of independent random variables. Multivariate change of variables. The Jacobian. Examples. Reading assignment: 4.6.

• Week Eight.

  During this week, we will look at inequalities. We will cover several inequalities that prove useful for a variety of purposes. For now, view these as tools which will be used later in the year.

• Lecture 21. Monday, November 9. Multivariate random vectors.

  Continuation of work on multivariate change of variables.

• Veteran's Day. Wednesday, November 11.

• Lecture 22. Friday, November 13. Inequalities.

  The Cauchy-Schwarz inequality. Jensen's inequality. Reading assignment: 4.7, 4.9.

• Week Nine.

  This week, we will begin with the notion of a random sample. From here, we will examine properties of the sampling distributions of a pair of statistics--the sample mean and the sample variance. Remarkably, when the random sample is drawn from a normal distribution, these statistics are independent of one another. These statistics lead to three distributions of great importance: the chi-squared distribution, the t distibution, and the F distribution.

• Lecture 23. Monday, November 16. Random samples.

  The random sample. The simple random sample. Statistics and sampling distributions. The sample mean, the sample variance. Convolutions. Reading assignment: 5.1, 5.2.

• Lecture 24. Wednesday, November 18. Normal distributions.

  Normal distributions. Independence of the sample mean and sample variance. Quiz II. Reading assignment: 5.3.1.

• Lecture 25. Friday, November 20. Distributions derived from the normal.

  The chi-squared distribution. The t distribution. The F distribution. Reading assignment: 5.3.2.

• Week Ten.

  The tenth week of the course is a short one, limited to two lectures. We will work a bit on order statistics and begin the study of convergence.

• Lecture 26. Monday, November 23. Order statistics.

  Order statistics. The joint pdf of the order statistics (via a change of variables). Marginal and partially marginal pdfs of order statistics. Order statistics of the uniform distribution. Reading assignment: 5.4.

• Lecture 27. Wednesday, November 25. Convergence in probability.

  Convergence in probability. Almost sure convergence. Contrasting the two types of convergence. Reading assignment: 5.5.1.

• Thanksgiving Holiday. Friday, November 27.

  No class today, the university is closed.

• Week Eleven.

  Oddly, the eleventh week of the ten week quarter. This week will be devoted to the study of asymptotics. We will continue our study of convergence, covering several different types of convergence, working through examples and seeing the relationships between them. We will also learn about a marvelous method for making asymptotic inference called the delta method.

• Lecture 28. Monday, November 30. Convergence in distribution.

  The Central Limit Theorem. Slutsky's Theorem. Convergence in quadratic mean. Reading assignment: 5.5.3.

• Lecture 29. Wednesday, December 2. The delta method.

  Asymptotic normality. Asymptotic normality and transformations. Examples. Reading assignment: 5.5.4.

• Lecture 30. Friday, December 4. Course wrap-up.

  More work on convergence. Worked examples. Reading assignment: None!

Course documents

• The course syllabus

• The first homework assignment - Due Monday, September 28
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw1soln.pdf

• The second homework assignment - Due Monday, October 5
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw2soln.pdf

• The third homework assignment - Due Monday, October 12
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw3soln.pdf

• The fourth homework assignment - Due Monday, October 19
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw4soln.pdf

• The fifth homework assignment - Due Monday, October 26
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw5soln.pdf

• The sixth homework assignment - Due Wednesday, November 4
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw6soln.pdf

• The seventh homework assignment - Due Friday, November 13
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw7soln.pdf

• The eighth homework assignment - Due Monday, November 23
• Solutions are available at http://www.stat.osu.edu/~snm/620/hw8soln.pdf

• The ninth homework assignment - Tentatively due Monday, November 30

• The tenth homework assignment - Due Friday, December 4 !!This assignment may be adjusted!!

Additional resources

• Cramer, H. (1946). Mathematical Methods of Statistics. Princeton: Princeton University Press. This one is a classic. It's been reprinted recently, in paperback form.
• DeGroot, M.H. (1986). Probability and Statistics, 2nd ed.. New York: Addison-Wesley. There is a revised version of this book, with Mark Schervish as a co-author.
• Lavine, M.L. (continuously updated). Introduction to Statistical Thought is an on-line book that can be accessed from the author's home page.
• Lindgren, B. (1976). Statistical Theory, 3rd ed. New York: Macmillan. Another classic. The written descriptions of mathematical concepts are exactly right in this book.