Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley
Office Hours: Mondays 4:00-5:00, Wednesdays 3:00-5:00, Fridays 11:00-12:00; additional availability by appointment
Textbook: Probability with Applications and R, by Robert P. Dobrow and Amy S. Wagaman, ISBN: 1119692385;
available as a free e-text
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.
- General information. A selection of problems will be assigned to be written up individually and turned in each week.
- These problems will be due Fridays at 5 pm.
- You may work with others but the writing should be done on your own.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on Fridays at 5 pm.
- Please resubmit (only the problems you’re revising) on Gradescope.
| Assignment | Due |
|---|---|
| Homework 0 | Jan 26 |
| Homework 1 | Feb 2 |
| Homework 2 | Feb 9 |
| Homework 3 | Feb 16 |
| Homework 4 | Feb 23 |
| Homework 5 | Mar 8 |
| Homework 6 | Mar 15 |
| Exam 1 revisions | Mar 29 |
| Homework 7 | Mar 29 |
| Homework 8 | Apr 12 |
| Homework 9 | Apr 19 |
| Homework 10 | Apr 26 |
| Exam 2 revisions | May 3 |
Quizzes
There will be quizzes most weeks that will be given on Fridays. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Problems will always be related to the previous homework and class topics.
| Quiz | Date | Material |
|---|---|---|
| Quiz 1 | Feb 2 | Homework 0 |
| Quiz 2 | Feb 9 | Homework 1 |
| Quiz 3 | Feb 16 | Homework 2 |
| Quiz 4 | Feb 23 | Homework 3 |
| Quiz 5 | Mar 15 | Homework 5 |
| Quiz 6 | Mar 29 | Homework 6 |
| Quiz 7 | Apr 19 | Homework 8 |
| Quiz 8 | Apr 26 | CLT activity |
Exams
There will be three exams. The dates for the exams are subject to change slightly.
| Exam | Due Date | Format | Material |
|---|---|---|---|
| Exam 1 | Mar 1 | in-class | Weeks 1-4 |
| Exam 2 | Apr 5 | in-class | Weeks 5-8 |
| Exam 3 | May 3-7 | self-scheduled | Weeks 10-14 |
Course plan
Our plan is to cover most of chapters 1-9 in the textbook, with some sections omitted due to time constraints. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.
Chapter 1
Tuesday
- Topic: Sections 1.1-1.3: Introduction. We discuss some basic terminology (sample space, events, outcomes) and introduce the notion of a probability function.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet. If you have time, try Exercises 1.6 and 1.8 at the end of Chapter 1 and read the first two pages of Section 1.4.
Thursday
- Topic: Sections 1.4: Properties of probabilities. We discuss set operations (union, intersection, complement) and derive general properties of probability functions, including the classical inclusion-exclusion formula for the union of two events.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet. If you have time, try Exercises 1.31, 1.32, 1.33 at the end of Chapter 1 and read Section 1.5.
Friday
- Topic: Section 1.5-1.6: Counting I. We discuss the multiplication principle and how to count outcomes in random experiments involving ordered sequences.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and work on Homework 1. If you have time, read the first two pages of Section 1.7.
Chapter 1
Tuesday
- Topic: Section 1.7: Counting II. We discuss counting unordered subsets. In other words, we are interested in counting outcomes in experiments where order does not matter.
- Class materials: Lecture notes, worksheet
- After class: Make sure to bring a laptop that can run R to our next class. Finish today’s worksheet and work on Homework 0 redos and Homework 1.
Thursday
- Topic: Section 1.9: Introduction to simulation. We introduce the notion of Monte Carlo algorithms for approximating the probability of an event. We also discuss some basic R techniques and implement our first Monte Carlo simulation.
- Class materials: Lecture notes, Markdown worksheet, pdf output, solutions
- After class: Finish today’s R lab and work on Homework 0 redos and Homework 1. Study for Friday’s quiz, which covers material from Homework 0 (Sections 1.1-1.3). Read Examples 1.29 and 1.31 in Section 1.8.
Friday
- Topic: Section 1.8: Complements and inclusion-exclusion. We discuss problem-solving strategies for computing the probability that at least one event occurs.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and work on Homework 2. Read the first three pages of Chapter 2.
Chapter 2
Tuesday
- Topic: Sections 2.1, 2.3: Introduction to conditional probability. We introduce the definition of conditional probability and discuss a technique for computing probabilities of sequential events.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 2.6, 2.7, 2.11. Read up through Example 2.10 in Section 2.4.
Thursday
- Topic: Sections 2.4, 2.5: Law of total probability and Bayes’ rule. We discuss the idea that the probability of an event can be computed by conditioning on an auxiliary event. We also discuss the idea of inverting conditional probability.
- Class materials: Lecture notes, worksheet
- After class: Study for Friday’s quiz, which covers material from Homework 1 (Sections 1.4-1.6). Read the first two pages of Chapter 3.
Friday
- Topic: Sections 2.6, 3.1-3.2: Independence and random variables. We discuss what it means for events to be independent. Along side that we introduce the concept of a random variable.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 3 and read Section 3.3.
Chapter 3
Tuesday
- Topic: Section 3.4: Binomial distribution. We introduce the Bernoulli and binomial distributions and discuss the kinds of quantities that such random variables can be used to model.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and try Exercises 3.14 and 3.15. Read the first page of Section 3.5.
Thursday
- Topic: Section 3.5: Poisson distribution. We discuss the idea of counting arrivals over a fixed time period and derive the Poisson distribution is a limit of the binomial distribution.
- Class materials: Lecture notes, worksheet
- After class: Study for Friday’s quiz, which covers material from Homework 2 (Sections 1.7-1.8).
Friday
- Topic: Sections 4.1-4.2: Expectation. We introduce the notion of the expected value of a random variable and derive formulas for the expectation of the uniform distribution and Poisson distribution.
- Class materials: Lecture notes, worksheet, Visualization of distributions, Rmd file
- After class: Work on Homework 4 and read the first two pages of Section 4.3.
Chapter 4
Tuesday
- Topic: Section 4.3: Joint distributions. We discuss probability mass functions of two variables in order to describe probabilities involving multiple random variables.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 4.17, 4.18, 4.19 and read the first three pages of Section 4.6.
Thursday
- Topic: Sections 4.5-4.6: Linearity of expectation and variance. We introduce the notion of variance of a random variable and how it is used to give further information about a probability distribution. We also discuss how to compute the expectation and variance of linear combinations of random variables.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 5.1 and study for tomorrow’s quiz which covers material from Homework 3 (Sections 2.1-2.5).
Friday
- Topic: Sections 5.1, 5.3: Geometric and negative binomial distributions. We discuss two distributions which count the number of trials until a desired number of successes occur in repeated independent Bernoulli trials.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 5 and begin studying for our first exam. Review lecture notes and try some suggested exercises from Homeworks 1-4.
Chapter 5
Tuesday
- Topic: Sections 5.4: Hypergeometric distribution. We discuss a distribution used in modeling counting for random experiments that involve sampling without replacement.
- Class materials: Lecture notes, worksheet
- After class: Study for Exam 1.
Thursday
- Topic: Review. We spend the day reviewing for Exam 1.
- Class materials: Review worksheet, solutions
- After class: Study for Exam 1.
Friday
- Topic: Exam 1.
- After class: Work on Homework 5 and read the first two pages of Section 5.2.
Chapter 6
Tuesday
- Topic: Section 5.2: Moment generating functions. We discuss a function that is useful for computing moments of a random variable as well as characterizing the distribution of independent sums of random variables. We will continue the discussion of this function later in the semester when proving the Central Limit Theorem.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 5.13, 5.14, and 5.15. Read Section 6.1 up through Example 6.2.
Thursday
- Topic: Section 6.1: Probability density functions. We introduce the notion of continuous random variables and how probabilities of events involving a continuous random variable can be computed using the integral of a so-called the probability density function.
- Class materials: Lecture notes, worksheet, \(u\)-substitution review, improper integration review, calculus review problems
- After class: Try Exercises 6.3 and 6.4. Read Section 6.2 up through Example 6.6.
Friday
- Topic: Section 6.2: Cumulative distribution functions. We discuss a particular antiderivative of a probability density function called the cumulative distribution function.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 6 and skim through Section 6.3.
Chapter 6
Tuesday
- Topic: Section 6.3, 6.4: Expectation and variance, uniform distribution. We discuss how to compute the mean and variance of a continuous random variable. We also introduce the continuous uniform distribution.
- Class materials: Lecture notes, worksheet
- After class: Read Section 6.5 up through Example 6.13.
Thursday
- Topic: Sections 6.5: Exponential distribution. We introduce the exponential distribution, which gives a model for waiting times or time until an arrival in a Poisson process.
- Class materials: Lecture notes, worksheet
- After class: Read Section 6.6 up through Example 6.15.
Friday
- Topic: Section 6.6: Joint distributions. We introduce the notion of the joint density of two random variables and how probabilities involving two continuous random variables can be computed by double integrals of the joint density. The main topic of discussion is a review of setting up double integrals.
- Class materials: Lecture notes, worksheet
- After class: Start working on Homework 7, but make sure to enjoy spring break!
Spring Break
Tuesday
- Topic: Spring break, no class.
Thursday
- Topic: Spring break, no class.
Friday
- Topic: Spring break, no class.
Chapter 6
Tuesday
- Topic: Section 6.6: Joint distributions. We continue our discussion of joint densities, getting more practice with multivariate integration. We also discuss the expected value of a function of two random variables.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 6.25, 6.26, and 6.28. Read page 254 on marginal densities.
Thursday
- Topic: Section 6.7: Independence. We discuss marginal densities and what it means for two continuous random variables to be independent.
- Class materials: Lecture notes, worksheet
- After class: Try Exercises 6.29 and 6.30. Read the first three pages of Section 7.1.
Friday
- Topic: Section 7.1: Normal distribution. We discuss some introductory properties of the normal distribution.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 8 and begin studying for our first exam. Review lecture notes and try some suggested exercises from Homeworks 5-7.
Chapter 7
Tuesday
- Topic: Sections 7.1: Normal distribution. We discuss the moment generating function of a normal random variable and study independent sums of normal random variables.
- Class materials: Lecture notes, worksheet
- After class: Study for Exam 2.
Thursday
- Topic: Review. We spend the day reviewing for Exam 2.
- Class materials: worksheet, solutions,
- After class: Study for Exam 2.
Friday
- Topic: Exam 2.
- After class: Work on Homework 8 and read the first two pages of Section 8.1.
Chapter 8
Tuesday
- Topic: Community day, no class.
Thursday
- Topic: Section 8.1: Functions of random variables. Given a random variable \(X\), we discuss how to find the density of a new random variable \(Y=g(X)\) given by a transformation of \(X\) when \(g\) is an invertible function.
- Class materials: Lecture notes, worksheet
- After class: Read the first page of Section 8.2.
Friday
- Topic: Section 8.2: Min and max of random variables. We discuss finding the distributions of the minimum and maximum of a finite collection of independent random variables.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 9.1 and start working on Homework 9.
Chapter 9
Tuesday
- Topic: Sections 9.1: Conditional densities. We discuss computing conditional probabilities involving two continuous random variables. Along the way we introduce a continuous version of Bayes’ formula.
- Class materials: Lecture notes, worksheet
- After class: Read the first two pages of Section 9.3.
Thursday
- Topic: Section 9.3: Conditional expectation. We discuss how to compute conditional expectations in both the discrete and continuous random variable cases.
- Class materials: Lecture notes, worksheet
- After class: Read pages 372 and 373.
Friday
- Topic: Section 9.3: Conditional expectation. We conclude our discussion of conditional expectations by discussing the notion of conditional expectation as a random variable and the Law of Total Expectation.
- Class materials: Lecture notes, worksheet
- After class: Read the first three pages of Chapter 10 and start working on Homework 10.
Chapter 10
Tuesday
- Topic: Sections 10.1-10.2, 10.4: Law of large numbers and Monte Carlo integration. We discuss our first main probabilistic limit theorem, the Law of Large Numbers, and how it can be used as a numerical integration algorithm.
- Class materials: Lecture notes, worksheet
- After class: Read the first four pages of Section 10.5.
Thursday
- Topic: Section 10.5. Central Limit Theorem. We discuss our second main probabilistic limit theorem, the Central Limit Theorem, how it can be viewed as a refinement of the Law of Large Numbers, and how it is used to approximate probabilities in various examples.
- Class materials: Lecture notes, worksheet
- After class: Work on and finish Homework 10.
Friday
- Topic: Sections 10.6: Central limit theorem. We discuss the proof of the CLT.
- Class materials: Lecture notes, worksheet
- After class: Begin studying for Exam 3 and take a moment to relax before finals!
Getting help
Here are a few ways to get help:
- Office hours: Mondays 4:00-5:00, Wednesdays 3:00-5:00, Fridays 11:00-12:00; additional availability by appointment
- Evening help: Our class has two TAs who will be holding evening help. Details will be posted on Moodle.
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what you know on homework in order to get good feedback.
- Piazza: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum regularly and answer questions as they come up, but my hope is everyone in the class will pitch in and answer others’ questions as another form of participation in the class.
Resources
- Everyone is invited to join DataCamp, which provides an introductory R tutorial. It’s a convenient way to gain some familiarity with R, a useful tool for our course and beyond. Our textbook also provides a thorough tutorial of some R basics in the appendix.
- Our textbook also has a useful collection of R scripts available; contained there are all the R code snippets you’ll notice interspersed in the text.
- I’ve collected some resources to help you with some basics of RMarkdown.
- Here is a RMarkdown/LaTeX template file for writing nicely formatted documents, along with its pdf output.
- A LaTeX quick reference is available for commonly used symbols.
- RStudio Server is a cloud service that lets you edit and compile R and RMarkdown files through a web browser so that no local installation is needed. The server is hosted on the MHC network and you need to be on the VPN to access it if you’re away from campus.
- You can also install R and RStudio locally on your personal computer (you must install R before RStudio), or you can also use RStudio Cloud, which is a commercial RStudio cloud service with a free tier.
- There are many probability books out there. Reading our required textbook will be enough to get a strong understanding of the material in class, but if you’re curious, here are some other good books.
- A First Course in Probability by Sheldon Ross. Fundamentally similar in content as our textbook, but without the nuanced view toward real world applications and simulation. A big positive is that it has an immense number of exercises.
- Introduction to Probability by Dimitri Bertsekas and John Tsitsiklis. Similar to our textbook, but more barebones. Covers the important topics well, but it seems to miss the interesting detours and exercises that make probability come alive. It’s nice as a quick reference for the basics. There is also a great MIT open course based on the textbook given by one of its authors which has video lectures and notes.
- Adobe has a Merge PDF tool that is free to use over the web.