This is the web page of Math 1610 - Probability. You can view the syllabus of the course here. In this page I will post materials, announcements, and other relevant information, including the homework (that will also be announced in class). Recall that we have classes on Monday, Wednesday and Friday from 11:00am to 11:50am in Barus & Holley 160. Office hours are on Thursdays from 2:00pm to 4:00pm or by appointment.

Here is a link to the book that we are using in the course (Grinstead and Snell - Introduction to Probability). Here is the link to the webpage of the book, from which you can access many resources related to the content of the course. This includes solutions to many exercises in the book (in particular, the odd numbered ones). Those will sometimes be part of the homework but you will not need to turn them in, since you can check that you did them correctly yourselves.

Announcements

• (12/06/2017) All lecture notes uploaded.
• (11/27/2017) CORRECTION MADE: final exam correct date is December 20!
• (11/21/2017) My lecture notes up to -and not including- random walks are now available in the covered topics section of this page (below). Use at your own risk.
• (11/20/2017) Some solutions for the ninth homework.
• (11/5/2017) Office hours for the week of November 8 are moved to Tuesday 2pm-4pm.
• (11/5/2017) Solution to the pdf exercise in the seventh homework.
• (10/23/2017) Some solutions for the sixth homework.
• (10/22/2017) Extra office hours this week: Tuesday 24, 11am-1pm.
• (10/20/2017) Solution to exercise 2, section 4.3.
• (10/15/2017) Some solutions for the third homework.
• (09/22/2017) Solution to exercise 23, section 3.1.
• (09/20/2017) Some solutions for the first homework.
• (09/20/2017) Office hours for tomorrow are changed and are now 2pm-3pm and 4pm-5pm. Sorry!
• (09/20/2017) There will be no class on October 11. We will make up for it during reading period.
• (09/14/2017) The midterm will be in Barus & Holley 158 (this will be definitive when shopping period ends).
• (09/08/2017) The midterm date is confirmed: Thursday, October 26, 6-8pm.
• (09/07/2017) I will not be in Providence during the weeks of October 6 and October 13. However, classes continue normally during that period and someone else will give the lectures in my place.

Exams

• Midterm: Thursday, October 26, 6-8pm, Barus & Holley 158.
• Final: Wednesday, December 20, 9-12am, Smith-Buonanno Hall G12.

Homework (always to be handed in in class, or BEFORE the deadline in other ways that we may agree on). Dates in bold are the ones I will grade myself and which count more toward your grade. Dates in italics correspond to assignments that you do not need to hand in because they will consist of odd numbered problems from the book that you can check yourselves, or some other reason. Anyway, I strongly recommend that you do the exercises. They are part of the course.

• Friday, September 15: Section 1.2, exercises 2, 6, 8, 10, 14, 16, 18, 22, 26, 30. Some solutions.
• Friday, September 22: Section 1.2, exercises 23, 25, 31. Section 3.1, exercises 1, 5, 11, 13, 17, 23. Solution to exercise 23, section 3.1.
• Friday, September 29: Section 1.2, exercises 20, 28. Section 3.1, exercise 12. Section 3.2, exercises 8,12, 16, 22, 26, 34a. Some solutions.
• Friday, October 6: Section 3.3, exercise 1 (you need to read section 3.3 in order to do this one). Section 2.2, exercises 1, 5, 13, 15. Section 4.1, exercises 1, 7, 19, 25, 27. READ CHAPTER 5 OF THE BOOK.
• Friday, October 13: Section 4.1, exercises 16, 24, 30, 34, 44 (the last one requires reading a bit from the book). Section 4.2, exercises 4, 8. Section 4.3, exercise 2 (in both variants, 8 cards and 52 cards). Section 6.1, exercises 4, 5, 8. Solution to exercise 2, section 4.3.
• Friday, October 20: Section 6.1, exercises 16, 20a and b, 28, 34 (except b), 40. Section 6.2, exercises 4, 10, 12, 20. Section 6.3, exercises 10, 22, 28. Some solutions.
• Friday, October 27: No homework (midterm week).
• Friday, November 3: Section 7.2, exercises 10, 19, 20. Section 8.1, exercises 8, 12. Section 8.2, exercises 4, 12. Plus the exercise in this pdf. Solution to the exercise in the pdf.
• Friday, November 10: Section 8.1, exercise 17 (you don't need to look at exercise 15, since we have a statement of the strong law of large numbers as we saw it in class). Section 8.2, exercises 9, 17. Section 9.1, exercises 3, 5, 9, 17 (for the last one you may need to read a bit from the book).
• Friday, November 17: Section 5.2, exercise 38. Section 8.1, exercise 16. Section 8.2, exercise 18 (see the previous one). Section 9.1, exercises 4, 10. Section 9.2, exercises 2, 9 (weak law of large numbers, not the strong one). Section 9.3, exercise 14. Some solutions.
• Friday, November 24: No homework (Thanksgiving).
• Friday, December 1: Section 10.1, exercises 3, 4, 10. Section 10.2, exercises 4,6. Section 10.3, exercises 6, 8.
• Friday, December 8: Section 12.1, exercises 5, 13, 14, 15.

Summary of topics that we have covered in class (will be updated approximately after each week):

• Week of September 6: Notion of sample space, probability distribution function, the birthday problem, basic properties of probabilities, infinite sample spaces (section 1.2 of the book). Notes.
• Week of September 13: Combinatorics, including permutations, variations and combinations, Stirling's formula, binomial theorem, statement of the inclusion-exclusion principle (sections 3.1 and 3.2 of the book, although we gave different proofs in a couple places; also, we skipped one or two things). Notes.
• Week of September 20: Proof of the inclusion-exclusion principle, probability that a permutation does not have fixed points, continuous random variables, density and cumulative distribution functions, discrete experiments that become continuous. Notes.
• Week of September 27: Conditional probability, notion of independence, independence of more than two sets, independence of random variables, independent trials. Notes.
• Week of October 4: Bayes formula, expectation of a random variable, properties. Notes.
• Week of October 11: Variance of random variables. Variance of the sum of random variables. Notes.
• Week of October 18: Conditional expectation, sums of independent random variables, convolutions. Notes.
• Week of October 25: Chebyshev's inequality, weak law of large numbers, random variables as functions. Solutions of the midterm. Notes.
• Week of November 1: Strong law of large numbers (without proof), conditional expectation for random variables viewed as functions, central limit theorem for sums of independent Bernoulli random variables, more general statements of the central limit theorem. Notes.
• Week of November 8: Moment generating functions. Properties of moment generating functions. Moment generating functions determine probability distributions. Ordinary generating functions. Notes.
• Week of November 15: Moment generating function of the Gaussian. Proof of the central limit theorem. Branching processes and the problem of extinction. Introduction to discrete random walks. Notes.
• Week of November 22: Some probabilities associated to one dimensional discrete random walks. Notes.
• Week of November 29: Two dimensional discrete random walks. Higher dimensional discrete random walks. Notes.
• Week of December 6: Expected number of returns to the origin in a one dimensional random walk. Gambler's ruin.