Chapter 1

Tuesday

• Topic: Convocation Day, no class.

Thursday

• 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.

Friday

• 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. Post a message on the class [online forum][] introducing yourself. Some things you might want to include are: your name, pronouns, what classes you are taking, and what are some things you like to do outside of academics. Please create your own introduction post and feel free to read and reply to your classmates’ posts.

Chapter 1

Tuesday

• 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.

Thursday

• 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: Read Examples 1.29 and 1.31 in Section 1.8. Finish today’s worksheet and work on Homework 0 redos and Homework 1. Study for Friday’s quiz, which will have a question about sample spaces, events, and equally likely outcomes and a question about the three defining properties of a probability function.

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: Make sure to bring a laptop that can run R to our next class and start on Homework 2.

Chapter 2

Tuesday

• 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 continue to work on Homework 2. Read the first three pages of Chapter 2.

Thursday

• Topic: Mountain Day, no class.
• After class: Study for tomorrow’s quiz, which will focus on Venn diagrams, expressing events using set operations (union, intersection, complement), and using the inclusion-exclusion formula for two events.

Friday

• 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.

Chapters 2, 3

Tuesday

• 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: Read the first two pages of Chapter 3.

Thursday

• 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: Read Section 3.3. Study for tomorrow’s quiz, which will focus on counting, both when order does and does not matter, and using the inclusion-exclusion formula for three sets.

Friday

• 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.

Chapter 4

Tuesday

• 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: Finish today’s worksheet and work on this week’s homework.

Thursday

• 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: Read the first two pages of Section 5.1. Study for tomorrow’s quiz. It will have two questions on conditional probability. Make sure you know the defining formula for \(P(A\mid B)\) and how to identify when a problem is asking you to compute a conditional probability.

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: Read the first two pages of Section 4.3.

Chapter 5

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: 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. Begin studying for Exam 1. Enjoy your break!

Exam

Tuesday

• Topic: Fall break, no class.

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: Read Section 6.1 up through Example 6.2.

Chapter 6

Tuesday

• Topic: Section 5.2: Moment generating functions. We continue our discussion of moment generating functions that was started last time.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Chapter 6.

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. Study for your quiz tomorrow. It will involve computing expectations involving one variable and computing probabilities involving the Poisson, geometric, and negative binomial distributions.

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 7 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. For tomorrow’s quiz, you should know how to find a moment generating function and use it to compute \(E[X]\) and \(E[X^2]\). You should also be able to express a joint probability mass function as a table, compute marginal distributions, and compute expectations that involve two variables using a double summation.

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: Get started on Homework 8.

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. Study for this week’s quiz. I want you to know the defining properties of a probability density function, how to find the cumulative distribution function of a continuous random variable given its density and vice versa, and how to compute probabilities and expectations of continuous random variables.

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 9 and begin studying for our next 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: Continue studying 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: Read the first two pages of Section 9.1 and continue working on Homework 9.

Chapter 8

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. For tomorrow’s quiz, I want you to know how to set up double integrals for joint probabilities using either order of integration, as well as the density and CDF of the exponential distribution and the memoryless property.

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. Work on Homework 10.

Chapter 9

Tuesday

• Topic: November break, no class. I will hold office hours during class time and the usual 4-5 pm office hour is cancelled.
• After class: Enjoy the break!

Thursday

• Topic: November break, no class.

Friday

• Topic: November break, no class.

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: Finish the quiz, begin studying for Exam 3, and take a moment to relax before finals!

Review

Tuesday

• Topic: Review day.
• Class materials: worksheet, solutions
• After class: Submit Quiz 10 on Gradescope by 5 pm and study for Exam 3! Enjoy your winter break! Keep in touch!