**Course: Math 407, Probability Theory I, Spring 2023, Section 39975R**

**Prerequisite:** MATH 226 or MATH 227 or MATH 229

**Course Content:** Probability spaces, discrete and continuous distributions, moments, characteristic functions, sequences of random variables, laws of large numbers, central limit theorem, special probability laws.

*Last update:* 26 October 2023

**Instructor:** Steven Heilman, stevenmheilman(@-symbol)gmail.com

**Office Hours:** 8AM-10AM, Tuesdays, on zoom [link posted on blackboard]

**Lecture Meeting Time/Location:** Mondays, Wednesdays, and Fridays, 10AM-1050AM, CPA 152

**TA: **Quinn Le, ntle(@-symbol)usc.edu

**TA Office Hours: **held in the Math Center.

**Discussion Section Meeting Time/Location**:

- 39976R, Tuesdays and Thursdays, 2PM-250PM, CPA 102
- 39977R, Tuesdays and Thursdays, 3PM-350PM, CPA 102

**TA Office Hours:** Occur in the Math Center.

**You are not required to buy a textbook**. Free lecture notes are provided below.

**Recommended Textbook:** D. P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, 2nd edition. (The book is freely available online)

**Another Recommended Textbook:** Sheldon Ross, A First Course in Probability, any edition. (The book is freely available online)

**Another Recommended Textbook:** __Elementary Probability for Applications__, Durrett.

**First Midterm:** Friday, February 10, 10AM-1050AM, CPA 152

**Second Midterm:** Friday, March 24, 10AM-1050AM, CPA 152

**Final Exam:** Monday, May 8, 8AM-10AM, CPA 152

**Other Resources:** Supplemental Problems from the textbook. An introduction to mathematical arguments, Michael Hutchings, An Introduction to Proofs, How to Write Mathematical Arguments

**Email Policy:**

- My email address for this course is stevenmheilman@gmail.com
- It is your responsibility to make sure you are receiving emails from stevenmheilman@gmail.com , and they are not being sent to your spam folder.
- Do NOT email me with questions that can be answered from this website.

**Exam Procedures**: Students must bring their USCID cards to the midterms and to the final exam. Phones must be turned off. Cheating on an exam results in a score of zero on that exam. Exams can be regraded at most 15 days after the date of the exam. This policy extends to homeworks as well. All students are expected to be familiar with the USC Student Conduct Code. (See also here.)

Accessibility Services: If you are registered with accessibility services, I would be happy to discuss this at the beginning of the course. Any student requesting accommodations based on a disability is required to register with Accessibility Services (OSAS) each semester. A letter of verification for approved accommodations can be obtained from OSAS. Please be sure the letter is delivered to me as early in the semester as possible. OSAS is located in 301 STU and is open 8:30am-5:00pm, Monday through Friday.

https://osas.usc.edu/

213-740-0776 (phone)

213-740-6948 (TDD only)

213-740-8216 (fax)

OSASFrontDesk@usc.edu

Discrimination, sexual assault, and harassment are not tolerated by the university. You are encouraged to report any incidents to the *Office of Equity and Diversity* http://equity.usc.edu/ or to the *Department of Public Safety* http://capsnet.usc.edu/department/department-public-safety/online-forms/contact-us. This is important for the safety whole USC community. Another member of the university community - such as a friend, classmate, advisor, or faculty member - can help initiate the report, or can initiate the report on behalf of another person. *The Center for Women and Men* http://www.usc.edu/student-affairs/cwm/ provides 24/7 confidential support, and the sexual assault resource center webpage sarc@usc.edu describes reporting options and other resources.

**Exam Resources:** Here are the exams I used when I previously taught a similar course: Exam 1 Exam 1 Solution Exam 1v2 Exam 1v2 Solution Exam 2 Exam 2 Solution Exam 2v2 Exam 2v2 Solution Final Final Solution Finalv2 Finalv2 Solution Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions. Final Final Solutions. Here is a page containing old exams for another similar class. Here is a practice midterm (with solutions). Here are some more exams from a second quarter probability course: Exam 1 Exam 1 Solution Exam 2 Exam 2 Solution Final Final Solution. Exam 1 Exam 1 Solutions Exam 2 Exam 2 Solutions Final Final Solutions. Occasionally these exams will cover slightly different material than this class, or the material will be in a slightly different order, but generally, the concepts should be close if not identical.

**Homework Policy**:

- Homeworks are
**due at 2PM Thursdays**. - Homeworks are submitted in blackboard, under the "Assignments" tab. You are allowed unlimited submission "attempts" for an assignment, but only the last submission will be graded. To avoid internet issues, I recommend making your first submission of an assignment well in advance of the deadline. (Note that phone tethering can also give you an internet connection to a computer.)
- Homeworks should be submitted as single PDF documents. One way to create a PDF document from paper homework assignments is the freely available Adode Scan App
- Late homework is
**not accepted**. - If you still want to turn in your homework late, then the number of minutes late divided by ten will be deducted from the homework score. The exact deduction and rounding procedure is not guaranteed to be accurate.
- The
**lowest two**homework scores will be dropped. This policy is meant to account for illnesses, emergencies, dropped internet connections, etc. - You may not use the internet to try to find answers to homework problems.
- Do not submit homework via email.
- Collaboration on homeworks is allowed and encouraged.
- All homework assignments must be written by you, i.e. you cannot copy someone else`s solution verbatim.

**Grading Policy:**

- The final course grade is weighted as the larger of the following two schemes.
- Scheme 1: class participation (3%), homework (22%), the first midterm (20%), the second midterm (20%), and the final (35%).
- Scheme 2: class participation (3%), homework (22%), largest midterm grade (30%), and the final (45%).

- If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
- Class participation is not the same as attendance. I will never explicitly take attendance, but I will notice if someone is frequently absent. Things that increase your class participation grade include: asking good questions, paying attention in class, showing up on time or early to class, etc. Things that decrease your class participation grade include: excessive talking or disruptions during class, frequent absences, excessive texting/smartphone usage in class, frequent tardiness, etc.
- You must take the final exam to pass the course.

**Tentative Schedule**: (This schedule may change slightly during the course.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

1 | Jan 9: 1.1, Sets | Jan 11: 1.2, Probabilistic Models | Jan 12: Homework 0 due (ungraded) | Jan 13: 1.2, Probabilistic Models | |

2 | Jan 16: No class: | Jan 18: 1.3, Conditional Probability | Jan 19: Homework 1 due | Jan 20: 1.3, Conditional Probability | |

3 | Jan 23: 1.4, Total Probability Theorem and Bayes' Rule | Jan 25: 1.5, Independence | Jan 26: Homework 2 due | Jan 27: 1.5, Independence | |

4 | Jan 30: 1.6, Counting | Feb 1: 2.1, Discrete Random Variables | Feb 2: Homework 3 due | Feb 3: 2.2, Probability Mass Function | |

5 | Feb 6: 2.3, Functions of Random Variables | Feb 8: 2.4, Expectation and Variance | Feb 9: No homework due | Feb 10: Midterm #1 | |

6 | Feb 13: 2.5, Joint PMFs, Covariance and Variance | Feb 15: 2.6, Conditioning | Feb 16: Homework 4 due | Feb 17: 2.6, Conditioning | |

7 | Feb 20: No class | Sep 30: 2.7, Independence | Feb 23: Homework 5 due | Feb 24: 2.7, Independence | |

8 | Feb 27: 3.1, Continuous random variables and PDFs | Mar 1: 3.1, Continuous random variables and PDFs | Mar 2: Homework 6 due | Mar 3: 3.2, Cumulative Distribution Functions | |

9 | Mar 6: 3.3, Normal Random Variables | Mar 8: Joint PDFs of Multiple Random Variables | Mar 9: Homework 7 due | Mar 10: 3.5, Conditioning | |

10 | Mar 13: No class (spring break) | Mar 15: No class (spring break) | Mar 16: No class | Mar 17: No class (spring break) | |

11 | Mar 20: 3.5, Conditioning | Mar 22: 4.2, Covariance | Mar 23: No homework due. | Mar 24: Midterm #2 | |

12 | Mar 27: 4.4, Moment Generating Function | Mar 29: 4.4, Fourier Transform | Mar 30: Homework 8 due | Mar 31: 4.2 Convolution | |

13 | Apr 3: 7.1, Markov and Chebyshev Inequalities | Apr 5: 7.2, Weak Law of Large Numbers | Apr 6: Homework 9 due. | Apr 7: 7.3, Convergence in Probability | |

14 | Apr 10: 7.4, Central Limit Theorem | Apr 12: 7.4, Central Limit Theorem | Apr 13: Homework 10 due | Apr 14: 7.4, Central Limit Theorem | |

15 | Apr 17: 7.5, Strong Law of Large Numbers | Apr 19: 7.5, Strong Law of Large Numbers | Apr 20: Homework 11 due | Apr 21: 7.5, Strong Law of Large Numbers | |

16 | Apr 24: Leeway | Apr 26: Leeway | Apr 27: Homework 12 due | Apr 28: Leeway |

**Advice on succeeding in a math class:**

- Review the relevant course material
**before**you come to lecture. Consider reviewing course material a week or two before the semester starts. - When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
- Come to class with questions, so you can get more out of the lecture. Also, finish your homework at least
**two days**before it is due, to alleviate deadline stress. - Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, it might be beneficial to typeset your homework. Learning LaTeX is a good skill to have for doing mathematics. Here is a template .tex file if you want to get started typesetting.
- If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

**Homework**

**Homework .tex files**

**Exams**

**Supplementary Notes**