 ## Homework assignments (to be updated)

HW #0, due Friday, January 24 by 5pm: please send me an email introducing yourself: for instance, what name you prefer to be called, your major and hobbies, why you're interested in probability, or anything else you want to share or have questions about. It would also be helpful to attach a photo of yourself.

HW #1, due Friday, January 31 in class: Ch. 1 (pp. 15-17): 4, 5, 7, 8, 12, 18, 19, 20, 21, 24, 27, and Self-test problem 7 (p. 19); Ch. 2 (p. 48): 2, 3, 5 Answers

HW #2, due Friday, Feb 7 in class: Ch. 2 (pp. 48ff): 7, 8, 9, 12, 17, 18, 20, 21, 25, 27, 28, 32, 37 Answers

Some problem-solving tips from the book "How to Solve it" by Polya

HW #3, due Friday, Feb 14 in class: Ch. 2 (p. 51): 50, 53, and Ch. 3 (pp. 97ff): 1, 5, 6, 9, 10, 20, 23, 30, 47, 51, 56, 57, 59. Answers

HW #4, due Friday, Feb 21 in class: Ch. 3: 64, 66, 78, 81, 83, 84, and Ch. 4 (pp. 163ff): 1, 4, 5, 13, 14, 17, 19. The quiz will be graded as follows: if you do well, it will replace one of your low HW scores; if not, I'll drop the quiz. Answers

Quiz solutions

Old Midterm 1 with solutions

Midterm 1 covers homework problems up through and including HW #4 and lectures through Feb. 21. This means that the material in sections up through 4.2 and 4.10 will be examined more thoroughly than the material in sections 4.3-4.6 (for which you are responsible for the main facts and examples from lectures). Lecture material on 4.7-4.8 and homework problems on 4.3-4.6 will be on Midterm 2.
Midterm 1 solutions and Scan of pictures for Midterm 1

HW #5 due Friday, March 7 in class: Ch. 4 (pp. 163ff): 21, 23, 32, 35, 37, 38, 40, 42, 45, 48, 50. Answers

HW #6 due Friday, March 14 in class: Ch. 4: 55, 57, 59, 61, 63, 72, 73, 77, 78, 79, 84, 85. Ch. 5: 1, 2, 4, 5. Answers

HW #7 due Friday, March 21 in class: Ch. 5: 6, 10, 12, 13, 15, 18, 21, 22, 23, 25, 28, 32, 33, 34 Answers

HW #8 due Friday, April 4 in class: Ch. 5: 37, 38, 40, 41. Ch 6: 2, 7, 8, 9, 10. Answers

Midterm 2 covers homework problems up through and including HW #8 and lectures through Apr. 9. As you know, each midterm is comprehensive but emphasizes material not covered on Midterm 1, the earliest of which is lecture material on 4.7-4.8 and homework problems on 4.3-4.6.

Here are some study tips from Richard Laugesen:
• Make an outline with definitions and techniques from each section.
• Review the summary sheets for discrete and continuous distributions, their pmfs or pdfs, expectations, variances, and practical applications. (Hypergeometric and Cauchy are cool but not the most important.)
• After that review, you will benefit from re-working homework problems and in-class examples, because then you will have an improved mental framework to fit the problems into.
• Work through the practice exam, keeping in mind that our exam will be different.

HW #9 due Friday, April 18 in class: Midterm 2: #5 (definition of joint pdf), #6 (the pdf of X^2+Y^2 if X and Y are standard normal, hints below), and in the textbook: Ch. 6: 13, 14, 20, 21, 22, 23, 27, 28, 29, 33. Hints: In class we showed that if X is standard normal, then X^2 is Gamma(1/2, 1/2). We also discussed how Gamma(n,lambda) is the sum of n Exponential(lambda) random variables. There's a related fact that the sum of a Gamma(n,lambda) RV and a Gamma(m,lambda) RV is Gamma(n+m,lambda) assuming independence--why is this true? Or you can compute directly using polar coordinates: the cdf of Z = X^2 + Y^2 is P(X^2 + Y^2 < = R^2) = a double integral over the radius-R disc of the bivariate normal joint pdf discussed in class. Then take the derivative to get the pdf of Z, and you should recognize it. Answers Partial solutions for Midterm 2

HW #10 due Friday, April 25 in class: Ch. 6: 38, 40, 41, 42, 48, and Ch. 7: 5, 6, 7, 8, 11, 19, 21. Answers

What does randomness look like? The counterintuitive answer.

HW #11 due Friday, May 2 in class: Ch. 7: 30, 31, 33, 38, 39, 41, 42, 50, 51, 56, 57. Answers

Old final for studying. The final is cumulative: about one hour on old material and one hour on new material, though you will be allowed to work for the whole three hours. You may ask the TA to solve specific problems by emailing him at lincong2 (at) illinois (dot) edu.

## Schedule for lectures

Week 1: Introduction, sections 1.2, 1.3, 1.4
Week 2: Sections 1.5-1.6, 2.2-2.4
Week 3: Sections 2.5, 3.2-3.4
Week 4: Sections 3.4, 4.1, 4.2, 4.10
Week 5: Sections 4.3-4.6
Week 6: Sections 4.7, 4.8, review, Midterm 1 on February 28th
Week 7: Sections 4.8-4.9, 5.1-5.3
Week 8: Sections 5.4-5.7
Week 9: Sections 5.7, 6.1
Week 10: (Spring break--no class)
Week 11: Sections 6.2-6.5
Week 12: Section 6.5-6.6, review, Midterm 2 on April 11th
Week 13: Sections 7.2-7.3, 7.7
Week 14: Sections 7.4-7.5
Week 15: Sections 7.6-7.7, 8.2-8.3
Week 16: Section 8.3, review (last day of class: May 7)

## Syllabus

Introduction to mathematical probability: includes the calculus of probability, combinatorial analysis, random variables, expectation, distribution functions, moment-generating functions, and the central limit theorem. We will cover most of the material in the first eight chapters of the textbook.

## Why study probability?

There are two main reasons that probability is important: complexity and uncertainty. Some things are too complex to be analyzed exactly (like weather, the brain, social science), and probability provides a way of reducing this complexity and providing approximations that are useful.

Uncertainty is all around us and is usefully modeled as randomness: it appears in call centers, electronic circuits, quantum mechanics, medical treatment, epidemics, financial investments, insurance, games (both sports and gambling), online search engines, for starters. Probability is a good way of quantifying and discussing what we know about uncertain things, and then making decisions or estimating outcomes.

And the reason I study probability: statistical mechanics, which combines both the uncertainty of quantum mechanics and the complexity of zillions of particles interacting.

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