This is the web page of Math 1610 - Probability for the Fall of 2018. 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). Classes are on Tuesday and Thursday from 10:30am to 11:50am in Barus & Holley 165.

Main instructor: José Manuel Conde Alonso (myself).

• Email: jose_conde_alonso[at]brown[dot]edu
• Office hours: Tuesday 1-2pm, Wednesday 2-3pm or by appointment at my office (Kassar 214).

Problem solving fellow (teaching assistant): Kenneth Berglund.

• Email: kenneth_berglund[at]brown[dot]edu
• TA hours: Monday 8-9:30pm at Kassar House, room 105.

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

• (12-4-2018) Some solutions to homework 7 have been posted.
• (11-20-2018) Some solutions to homework 6 have been posted.
• (11-6-2018) Some solutions to homework 5 have been posted.
• (10-30-2018) Location of the final exam has been confirmed.
• (10-29-2018) Solutions (with rubrics) to the midterm have been posted Be careful when using them, I already found two mistakes in my solution!
• (10-22-2018) Some solutions to homework 4 have been posted.
• (10-16-2018) Some solutions to homework 3 have been posted.
• (9-28-2018) Some solutions to homework 2 (and from the previous week) have been posted.
• (9-21-2018) Date and time of the final are confirmed since shopping period ended.
• (9-19-2018) Extra office hours next week: Monday September 24, 11am-noon, Tuesday September 25, 2-3pm.
• (9-17-2018) Some solutions to homework 1 have been posted.
• (9-13-2018) Date, time and location of the midterm have been confirmed.

Exams

• Midterm: Wednesday, October 24, 6-8pm at Barus & Holley, room 168.
• Final: Monday, December 17, 2pm at Barus & Holley, room 165.

Homework (always to be handed in in class, or BEFORE the deadline in other ways that we may agree on). 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.

• Thursday, September 13: Section 1.2, exercises 2, 6, 8, 10, 14, 16, 18, 22, 26, 30 and the exercise in this pdf. Some solutions.
• Thursday, September 20: Section 1.2, exercises 23, 25, 31. Section 3.1, exercises 1, 5, 11, 13, 17, 23. Solution to problem 23, section 3.1.
• Thursday, September 27: Section 2.2, exercises 2, 6. Section 3.1, exercise 12. Section 3.2, exercises 8,12, 16, 22, 26, 34a and 36. And the exercise in this pdf. Some solutions.
• Thursday, October 4: Section 3.3, exercise 1 (you need to read section 3.3 in order to do this one). Section 2.2, exercises 1, 5, 15. Section 4.1, exercises 1, 7, 9, 13, 19, 25, 27.

Also, READ CHAPTER 5 OF THE BOOK (We will not be covering it specifically in class).

• Thursday, October 11: Section 4.1, exercises 8, 16, 24, 30, 44 (this one requires reading a bit from the book), 54. Section 4.2, exercises 4, 6, 8. Section 4.3, exercise 2 (only in the variant of 8 cards). Section 5.1, exercise 6. Section 5.2, exercise 18. And the exercise in this pdf. Some solutions.
• Thursday, October 18: Section 6.1, exercises 4, 5, 8, 16, 28, 34 (except b). Section 6.2, exercises 4, 10, 12, 20. Section 6.3, exercises 10, 28. And the exercise in this pdf. Some solutions.
• Thursday, October 25: No homework (midterm week).
• Thursday, November 1: Section 7.2, exercises 10, 19, 20. Section 8.1, exercises 8, 12. Section 8.2, exercises 4, 12. And the exercise in this pdf. Some solutions.
• Thursday, November 8: Section 8.2, exercises 9, 17. Section 9.1, exercises 3, 5, 9, 17. Section 9.3, exercise 13.
• Thursday, November 15: Section 8.2, exercises 17, 18. Section 9.1, exercises 4, 10. Section 9.2, exercises 2, 9. Section 9.3, exercise 14. Section 10.1, exercise 4. And the exercise in this pdf. Some solutions.
• Thursday, November 22: No homework (Thanksgiving).
• Thursday, November 29: Section 10.1, exercise 6. Section 10.2, exercises 4, 6, 8. Section 10.3, exercises 8, 10. Section 12.1, exercises 2, 6. And the exercise in this pdf. Some solutions.
• Thursday, December 6: Section 12.1, exercises 5, 13, 14, 15. Section 12.2, exercises 1, 4, 5.

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

• September 6: Notions of sample space, outcome, event, random variable, probability distribution function. Some properties.
• September 11: The birthday problem. Counting techniques: permutations, combinations.
• September 13: Stirling's formula. Formula for combinatorial numbers. Applications.
• September 18: Fixed points of permutations. Continuous random variables, density functions.
• September 20: Cumulative distribution functions.
• September 25: Conditional probability. Independence of two or more events.
• September 27: Independence of two random variables.
• October 2: Bayes formula, independent trial processes.
• October 4: Expectation. Existence and properties.
• October 9: The expectation is linear. Examples.
• October 11: Variance. Existence and properties.
• October 16: Conditional expectation. Sum of two independent discrete random variables.
• October 18: The variance as a minimizer. Sum of two independent continuous random variables. Examples.
• October 23: Convergence in probability. Chebyshev's inequality. Weak law of large numbers.
• October 25: Solutions of the midterm.
• October 30: Convergence in distribution The central limit theorem. Proof for Bernoulli trials.
• November 1: Computing probabilities with the central limit theorem. Moment generating functions. Moment generating function of a discrete, finite X determines distribution of X.
• November 6: Moment generating function of a continuous, bounded X determines distribution of X. Properties of moment generating functions. Moment generating function of a Gaussian.
• November 13: Proof of the central limit theorem. Ordinary generating functions.
• November 15: Generating function of a Cauchy random variable. Branching processes.
• November 20: Random walks. Introduction.
• November 27: Random walks in dimension 1. Pólya's theorem.
• November 29: Random walks in dimension 2.
• December 4: Random walks in dimension 3. Expected number of returns to the origin in dimension 1.
• December 6: Gambler's ruin.
• December 11: Review.