Math 464, Theory of Probability

ROLE:

Syllabus for Math 464, Section 002, Fall 2024

Location and Times

This course meets MWF 1-1:50pm in BIOW 210. The class will be also accessible by Zoom via a link provided by D2L.

Course Description

Course Prerequisites or Co-requisites

There are two ways to satisfy the requirements:

1. MATH 223 - Vector Calculus;
2. MATH 323 - Formal Mathematical Reasoning and Writing;

OR

1. MATH 223 - Vector Calculus;
2. MATH 313 - Introduction to Linear Algebra;
3. MATH/DATA 363 - Introduction to Statistical Methods.

If you think you have equivalent coursework, consider asking the instructor for permission to waive some requirements.

Instructor and Contact Information

Information	Data
Instructor	Professor Marek Rychlik
Office	Mathematics 605
Telephone	1-520-621-6865
Email	rychlik@email.arizona.edu
Instructor Homepage/Web Server	http://alamos.math.arizona.edu
Course Homepage	http://alamos.math.arizona.edu/math464
Course Homepage (Mirror)	http://marekrychlik.com/math464

Course Format and Teaching Methods

The course format is that of a conventional lecture (although delivered by Zoom), with in-class discussion and additional web-delivered content.

Course Objectives

1. Learn the introductory concepts of probability theory: chance experiments, probability spaces, probabilistic models.
2. Stochastic Independence.
3. Random variables: probability distributions, expectations and moments, functions of random variables, joint distributions, covariance and correlation, sums and moment generating functions, independence.
4. Limit theorems: Law of Large Numbers and Central Limit Theorem.
5. This course will prepare students for further courses in stochastic processes and mathematical statistics. It is a prerequisite for MATH 468 Applied Stochastic Processes and STAT 466 Theory of Statistics.

Learning outcomes

Upon completion of the course, the student will:

1. be able to set up and interpret probability models for a variety of chance experiments;
2. be able to interpret, explain, and apply probabilistic concepts such as probability, conditional probability, independence, expectations;
3. be able to "think probabilistically," e.g., break down long calculations of probabilities and/or expectations into simpler steps;
4. understand the relationship between random variables and their distributions / densities;
5. understand the meaning, scope, and consequences of the Law of Large Numbers and the Central Limit Theorem, including the meaning of convergence in distribution and convergence with probability 1;
6. be able to carry out concrete calculations with common distributions, e.g., the normal, exponential, and uniform distributions.

Generative AI use IS permitted or encouraged

In this course you are welcome and expected to use generative artificial intelligence/large language model tools, e.g. ChatGPT, Dall-e, Bard, Perplexity. Using these tools aligns with the course learning goals such as developing writing and programming skills, and ability to effectively use available information. Be aware that many AI companies collect information; do not enter confidential information as part of a prompt. LLMs may make up or hallucinate information. These tools may reflect misconceptions and biases of the data they were trained on and the human-written prompts used to steer them. You are responsible for checking facts, finding reliable sources for, and making a careful, critical examination of any work that you submit. Your use of AI tools or content must be acknowledged or cited. If you do not acknowledge or cite your use of an AI tool, what you submit will be considered a form of cheating or plagiarism. Please use the following guidelines for acknowledging/citing generative AI in your assignments:

• MLA
• APA

Absence and Class Participation Policy

Importance of attendance and class participation

Participating in course and attending lectures and other course events are vital to the learning process. As such, attendance is required at all lectures and discussion section meetings. Students who miss class due to illness or emergency are required to bring documentation from their healthcare provider or other relevant, professional third parties. Failure to submit third-party documentation will result in unexcused absences.

Missed Exams

Students are expected to be present for all exams. If a verifiable emergency arises which prevents you from taking an in-class exam at the regularly scheduled time, the instructor must be notified as soon as possible, and in any case, prior to the next regularly scheduled class. Make-up exams and quizzes will be administered only at the discretion of the instructor and only under extreme circumstances. If a student is allowed to make up a missed exam, (s)he must take it at a mutually arranged time. No further opportunities will be extended. Failure to contact your instructor as stated above or inability to produce sufficient evidence of a real emergency will result in a grade of zero on the exam. Other remedies, such as adjusting credit for other exams, may be considered.

COVID-19 related policies

As we enter the Fall semester, the health and wellbeing of everyone in this class is the highest priority. Accordingly, we are all required to follow the university guidelines on COVID-19 mitigation. Please visit http://www.covid19.arizona.edu for the latest guidance.

UA policies

The UA's policy concerning Class Attendance, Participation, and Administrative Drops is available at: http://catalog.arizona.edu/2015-16/policies/classatten.htm The UA policy regarding absences for any sincerely held religious belief, observance or practice will be accommodated where reasonable, http://policy.arizona.edu/human-resources/religious-accommodation-policy . Absences pre-approved by the UA Dean of Students (or Dean Designee) will be honored. See: http://uhap.web.arizona.edu/policy/appointed-personnel/7.04.02

Required Texts or Readings

Required Textbook

Probability : An Introduction. Geoffrey Grimmett, Dominic Welsh, and Professor of Mathematics (Retired) Dominic Welsh

Notes on exam administration

All examinations are planned to be administered during the class time, either in person or on Zoom.

If, due to unforseen circumstances, they cannot be held in person, they are held on Zoom using the "gallery view" mode.The exam papers for not in-person tests will be distributed on-line by D2L and collected electronically using D2L "dropbox" feature.

Exam/assignment listing with date and grade contribution

Exam or Assignment	Date	Grade contribution
Midterm 1	September 20 (Friday)	20%
Midterm 2	November 13 (Wednesday)	20%
Final Examination	December 16 (Monday), 1:00pm - 3:00pm	30%
Homework	See D2L	30%

Homework Assignments

Written homework consists of approximately twelve assignments equally contributing to the grade, each worth 30/12 = 2.5% of the grade. The assignment papers are provided at and collected via Gradescope, which is cloud-based software service integrated with D2L. The main function is to provide for semi-automatic grading. Things to keep in mind:

• The student will submit all graded work through Gradescope.
• Gradescope assignments, by their nature, are answer-oriented. The opportunity for partial credit is somewhat limited.
• Homework may not be submitted through e-mail, no exceptions.
• Due dates will be adjusted in Gradescope onlyas needed and the student is expected to check regularly for those adjustments. Due dates in D2L will not be updated.

Written homework is assigned regularly throughout the semester, for a total of approximately 120 problems (15 per covered chapter of the book). Two types of homework will be assigned:

1. Homework which consists of selected exercises in the required textbook.
2. Some custom homework will be composed by the instructor. Some of the custom problems will require programming.

Homework submission requirements

Using Gradescope for grading differs from other grading systems. Mainly, it uses AI to allow the instructor to accurately grade a larger number of problems than it would be possible otherwise. Some grading is completely automated (e.g., solutions to problems with a numerical answer). More comples answers may be grouped automatically by using Machine Learning, OCR and image analysis. However, it is possible to completely confuse the system by improperly structuring the submitted document. Therefore, please read the instructions below carefully and re-visit them as needed. Note that Gradescope supports automatic regrade requests which you can use if all fails.

The solutions must be structured in such a way that Gradescope can read them and that its 'AI' can interpret them. Your homework must be submitted as a PDF document, even if you use scanner or phone to capture images. Two typical workflows will be as follows:

1. Download the blank assignment (also called a 'template') from Gradescope.
2. Read and understand exactly what answers you need to provide. The space to enter the answer is a blue box, and marked with a label such as 'Q1.1' ("Question 1, part 1").
3. Work out the problem on "paper" (real or virtual), to obtain the answers. They must fit in the designated boxes in the 'template'. The size of the box is a hint from the instructor about the size of the answer (typically a number or a math formula) when entered by hand, using regular character size.
4. The recommended way to fill out the 'template' is paperless, by using suitable software and hardware (digital pen or tablet). I use a free program Xournal for this and it works great. You need to use it in combination with a digital pen or a tablet. It can produce a PDF easily, ready for submission to Gradescope.
5. You can also print the assignment on (real) paper, fill out the answers and scan the marked up document back to PDF format. However, the position of the boxes must be exactly (to a fraction of an inch) as in the original. Also, you may encounter a variety of "quality control" issues, especially if you are using a digital camera to scan the paper solution. All issues can be solved by a mix of the right hardware and software, but may not be the best time investment. The least troublesome way to scan is to use a real, flatbed scanner, e.g. in the library.
6. Upload the resulting document (a PDF of the 'template' marked up with your answers) to Gradescope. Your PDF must contain your name and student id in designated spaces. The Gradescope 'AI' will look for your name and student id, to properly associate it with your account.
7. After grading, the grade will be transmitted to D2L (Brightspace) and will be added to your 'Final Calculated Grade' automatically.
8. Do not reduce handwriting size! Reduce the size of your answer using
   • closed form expressions;
   • appropriate math functions, e.g., absolute value, min and max.
9. Under no circumstances write outside the provided space (boxes). Gradescope, and the grader only considers the content of the designated boxes.
10. IMPORTANT! Do not insert pages in the solution template. This will confuse Gradescope, and will result in reduced score and/or will require re-submission. However, you are encouraged to submit scratchwork. You should create pages at the end of the document. Similarly, if you run out of space in the template for your solution, you can continue the solution on a newly created page at the end of the document, adding a note in the template: "Solution continued on page 13" where page 13 will contain the continuation.

Programming and Software

The class will have small programming assignments. It is expected that you will be using software to gain insights into the assigned problems and subject matter. The programming assignments must be submitted in formats supported by Gradescope and the instructor. The number of programming languages will be limited two two or three. R will be supported and it is encouraged that you use it as it is most compatible with the course content.

For illustrating some aspects of the course, I will be using these programs (easy to download and free to use):

• A computer algebra system (CAS) called Maxima.
• The maxima GUI wxMaxima is a separate program and this is the preferred way to use Maxima if you are a beginner. During the previous semester students successfully applied Maxima to doing their homework, and I will provide examples of Maxima calculations in class.
• The R programming language and environment, with a rich set of tools to do simulations in probability and stochastic processes (and analyze data).
• The GUI (and more for R): RStudio. Makes use of R somewhat more pleasant, especially for a new user.

Final Examination

The final examination is scheduled for: December 16, 8:00am - 10:00am.

The time, data and general exam rules are set by the University and can be found at these links:

• Final Exam Regulations
• Final Exam Schedule

Grading Scale and Policies

The student in the class normally receives a letter grade A, B, C, D or E.

The cut-offs for the grades are:

Grade	% Range
A	90%+
B	80-90%
C	70-80%
D	60-70%
E	0-60%

Normally, individual tests and assignments will not be "curved". However, grade cut-offs may be lowered at the end of the semester (but not raised!) to reflect the difficulty of the assignments and other factors that may cause abnormal grade distribution.

The grade will be computed by D2L and the partial grade will be updated automatically by the system as soon as the individual grades are recorded.

General UA policy regarding grades and grading systems is available at https://catalog.arizona.edu/policy-type/grade-policies

Classroom Behavior Policy

To foster a positive learning environment, students and instructors have a shared responsibility. We want a safe, welcoming and inclusive environment where all of us feel comfortable with each other and where we can challenge ourselves to succeed. To that end, our focus is on the tasks at hand and not on extraneous activities (i.e. texting, chatting, reading a newspaper, making phone calls, web surfing, etc).

Threatening Behavior Policy

The UA Threatening Behavior by Students Policy prohibits threats of physical harm to any member of the University community, including to one's self. See: http://policy.arizona.edu/education-and-student-affairs/threatening-behavior-students .

Accessibility and Accommodations

Our goal in this classroom is that learning experiences be as accessible as possible. If you anticipate or experience physical or academic barriers based on disability, please let me know immediately so that we can discuss options. You are also welcome to contact Disability Resources (520-621-3268) to establish reasonable accommodations. For additional information on Disability Resources and reasonable accommodations, please visit http://drc.arizona.edu/ .

If you have reasonable accommodations, please plan to meet with me by appointment or during office hours to discuss accommodations and how my course requirements and activities may impact your ability to fully participate. Please be aware that the accessible table and chairs in this room should remain available for students who find that standard classroom seating is not usable. Code of Academic Integrity Required language: Students are encouraged to share intellectual views and discuss freely the principles and applications of course materials. However, graded work/exercises must be the product of independent effort unless otherwise instructed. Students are expected to adhere to the UA Code of Academic Integrity as described in the UA General Catalog. See: http://deanofstudents.arizona.edu/academic-integrity/students/academic-integrity http://deanofstudents.arizona.edu/codeofacademicintegrity .

UA Nondiscrimination and Anti-harassment Policy

The University is committed to creating and maintaining an environment free of discrimination, http://policy.arizona.edu/human-resources/nondiscrimination-and-anti-harassment-policy . Our classroom is a place where everyone is encouraged to express well-formed opinions and their reasons for those opinions. We also want to create a tolerant and open environment where such opinions can be expressed without resorting to bullying or discrimination of others.

Additional Resources for Students

UA Academic policies and procedures are available at: http://catalog.arizona.edu/2015-16/policies/aaindex.html Student Assistance and Advocacy information is available at: http://deanofstudents.arizona.edu/student-assistance/students/student-assistance

Confidentiality of Student Records

http://www.registrar.arizona.edu/ferpa/default.htm .

Subject to Change Statement

Information contained in the course syllabus, other than the grade and absence policy, may be subject to change with advance notice, as deemed appropriate by the instructor.

Significant Dates (from the Registrar's Website)

Registrar site

Material Covered

We will cover Parts A and B of the book ("probability theory up to the Central Limit Theorem"). Here is the approximate schedule with approximate dates when the particular sections shall be covered.

Id	PART	Chapter.Section	Title	Page	Date
1	A		BASIC PROBABILITY
2	A	1	Events and probabilities	3
3	A	1.1	Experiments with chance	3
4	A	1.2	Outcomes and events	3
5	A	1.3	Probabilities	6
6	A	1.4	Probability spaces	7
7	A	1.5	Discrete sample spaces	9
8	A	1.6	Conditional probabilities	11
9	A	1.7	Independent events	12
10	A	1.8	The partition theorem	14
11	A	1.9	Probability measures are continuous	16
12	A	1.10	Worked problems	17
13	A	1.11	Problems	19
14	A	2	Discrete random variables	23
15	A	2.1	Probability mass functions	23
16	A	2.2	Examples	26
17	A	2.3	Functions of discrete random variables	29
18	A	2.4	Expectation	30
19	A	2.5	Conditional expectation and the partition theorem	33
20	A	2.6	Problems	35
21	A	3	Multivariate discrete distributions and independence	38
22	A	3.1	Bivariate discrete distributions	38
23	A	3.2	Expectation in the multivariate case	40
24	A	3.3	Independence of discrete random variables	41
25	A	3.4	Sums of random variables	44
26	A	3.5	Indicator functions	45
27	A	3.6	Problems	47
28	A	4	Probability generating functions	50
29	A	4.1	Generating functions	50
30	A	4.2	Integer-valued random variables	51
31	A	4.3	Moments	54
32	A	4.4	Sums of independent random variables	56
33	A	4.5	Problems	58
34	A	5	Distribution functions and density functions	61
35	A	5.1	Distribution functions	61
36	A	5.2	Examples of distribution functions	64
37	A	5.3	Continuous random variables	65
38	A	5.4	Some common density functions	68
39	A	5.5	Functions of random variables	71
40	A	5.6	Expectations of continuous random variables	73
41	A	5.7	Geometrical probability	76
42	A	5.8	Problems	79
43	B		FURTHER PROBABILITY
44	B	6	Multivariate distributions and independence	83
45	B	6.1	Random vectors and independence	83
46	B	6.2	Joint density functions	85
47	B	6.3	Marginal density functions and independence	88
48	B	6.4	Sums of continuous random variables	91
49	B	6.5	Changes of variables	93
50	B	6.6	Conditional density functions	95
51	B	6.7	Expectations of continuous random variables	97
52	B	6.8	Bivariate normal distribution	100
53	B	6.9	Problems	102
54	B	7	Moments, and moment generating functions	108
55	B	7.1	A general note	108
56	B	7.2	Moments	111
57	B	7.3	Variance and covariance	113
58	B	7.4	Moment generating functions	117
59	B	7.5	Two inequalities	121
60	B	7.6	Characteristic functions	125
61	B	7.7	Problems	129
62	B	8	The main limit theorems	134
63	B	8.1	The law of averages	134
64	B	8.2	Chebyshev’s inequality and the weak law	136
65	B	8.3	The central limit theorem	139
66	B	8.4	Large deviations and Cramér’s theorem	42
67	B	8.5	Convergence in distribution, and characteristic functions	145
68	B	8.6	Problems	149
69	C		RANDOM PROCESSES
70	C	9	Branching processes	157
71	C	9.1	Random processes	157
72	C	9.2	A model for population growth	158
73	C	9.3	The generating-function method	159
74	C	9.4	An example	161
75	C	9.5	The probability of extinction	163
76	C	9.6	Problems	165
77	C	10	Random walks	167
78	C	10.1	One-dimensional random walks	167
79	C	10.2	Transition probabilities	168
80	C	10.3	Recurrence and transience of random walks	170
81	C	10.4	The Gambler’s Ruin Problem	173
82	C	10.5	Problems	177
83	C	11	Random processes in continuous time	181
84	C	11.1	Life at a telephone switchboard	181
85	C	11.2	Poisson processes	183
86	C	11.3	Inter-arrival times and the exponential distribution	187
87	C	11.4	Population growth, and the simple birth process	189
88	C	11.5	Birth and death processes	193
89	C	11.6	A simple queueing model	195
90	C	11.7	Problems	200
91	C	12	Markov chains	205
92	C	12.1	The Markov property	205
93	C	12.2	Transition probabilities	208
94	C	12.3	Class structure	212
95	C	12.4	Recurrence and transience	214
96	C	12.5	Random walks in one, two, and three dimensions	217
97	C	12.6	Hitting times and hitting probabilities	221
98	C	12.7	Stopping times and the strong Markov property	224
99	C	12.8	Classification of states	227
100	C	12.9	Invariant distributions	231
101	C	12.10	Convergence to equilibrium	235
102	C	12.11	Time reversal	240
103	C	12.12	Random walk on a graph	244
104	C	12.13	Problems	246
105	Appendix A		Elements of combinatorics	250
106	Appendix B		Difference equations	252
107	A		BASIC PROBABILITY
108	A	1	Events and probabilities	3
109	A	1.1	Experiments with chance	3
110	A	1.2	Outcomes and events	3
111	A	1.3	Probabilities	6
112	A	1.4	Probability spaces	7
113	A	1.5	Discrete sample spaces	9
114	A	1.6	Conditional probabilities	11
115	A	1.7	Independent events	12
116	A	1.8	The partition theorem	14
117	A	1.9	Probability measures are continuous	16
118	A	1.10	Worked problems	17
119	A	1.11	Problems	19
120	A	2	Discrete random variables	23
121	A	2.1	Probability mass functions	23
122	A	2.2	Examples	26
123	A	2.3	Functions of discrete random variables	29
124	A	2.4	Expectation	30
125	A	2.5	Conditional expectation and the partition theorem	33
126	A	2.6	Problems	35
127	A	3	Multivariate discrete distributions and independence	38
128	A	3.1	Bivariate discrete distributions	38
129	A	3.2	Expectation in the multivariate case	40
130	A	3.3	Independence of discrete random variables	41
131	A	3.4	Sums of random variables	44
132	A	3.5	Indicator functions	45
133	A	3.6	Problems	47
134	A	4	Probability generating functions	50
135	A	4.1	Generating functions	50
136	A	4.2	Integer-valued random variables	51
137	A	4.3	Moments	54
138	A	4.4	Sums of independent random variables	56
139	A	4.5	Problems	58
140	A	5	Distribution functions and density functions	61
141	A	5.1	Distribution functions	61
142	A	5.2	Examples of distribution functions	64
143	A	5.3	Continuous random variables	65
144	A	5.4	Some common density functions	68
145	A	5.5	Functions of random variables	71
146	A	5.6	Expectations of continuous random variables	73
147	A	5.7	Geometrical probability	76
148	A	5.8	Problems	79
149	B		FURTHER PROBABILITY
150	B	6	Multivariate distributions and independence	83
151	B	6.1	Random vectors and independence	83
152	B	6.2	Joint density functions	85
153	B	6.3	Marginal density functions and independence	88
154	B	6.4	Sums of continuous random variables	91
155	B	6.5	Changes of variables	93
156	B	6.6	Conditional density functions	95
157	B	6.7	Expectations of continuous random variables	97
158	B	6.8	Bivariate normal distribution	100
159	B	6.9	Problems	102
160	B	7	Moments, and moment generating functions	108
161	B	7.1	A general note	108
162	B	7.2	Moments	111
163	B	7.3	Variance and covariance	113
164	B	7.4	Moment generating functions	117
165	B	7.5	Two inequalities	121
166	B	7.6	Characteristic functions	125
167	B	7.7	Problems	129
168	B	8	The main limit theorems	134
169	B	8.1	The law of averages	134
170	B	8.2	Chebyshev’s inequality and the weak law	136
171	B	8.3	The central limit theorem	139
172	B	8.4	Large deviations and Cramér’s theorem	42
173	B	8.5	Convergence in distribution, and characteristic functions	145
174	B	8.6	Problems	149
175	C		RANDOM PROCESSES
176	C	9	Branching processes	157
177	C	9.1	Random processes	157
178	C	9.2	A model for population growth	158
179	C	9.3	The generating-function method	159
180	C	9.4	An example	161
181	C	9.5	The probability of extinction	163
182	C	9.6	Problems	165
183	C	10	Random walks	167
184	C	10.1	One-dimensional random walks	167
185	C	10.2	Transition probabilities	168
186	C	10.3	Recurrence and transience of random walks	170
187	C	10.4	The Gambler’s Ruin Problem	173
188	C	10.5	Problems	177
189	C	11	Random processes in continuous time	181
190	C	11.1	Life at a telephone switchboard	181
191	C	11.2	Poisson processes	183
192	C	11.3	Inter-arrival times and the exponential distribution	187
193	C	11.4	Population growth, and the simple birth process	189
194	C	11.5	Birth and death processes	193
195	C	11.6	A simple queueing model	195
196	C	11.7	Problems	200
197	C	12	Markov chains	205
198	C	12.1	The Markov property	205
199	C	12.2	Transition probabilities	208
200	C	12.3	Class structure	212
201	C	12.4	Recurrence and transience	214
202	C	12.5	Random walks in one, two, and three dimensions	217
203	C	12.6	Hitting times and hitting probabilities	221
204	C	12.7	Stopping times and the strong Markov property	224
205	C	12.8	Classification of states	227
206	C	12.9	Invariant distributions	231
207	C	12.10	Convergence to equilibrium	235
208	C	12.11	Time reversal	240
209	C	12.12	Random walk on a graph	244
210	C	12.13	Problems	246
211	Appendix A		Elements of combinatorics	250
212	Appendix B		Difference equations	252

Approximate timeline

Week	Dates	Topics Covered
1	Aug 26 - Aug 30	Introduction, Syllabus Review, Chapter 1: Events and probabilities (1.1 - 1.3)
2	Sep 2 - Sep 6	Labor Day (Sep 2, No Class), Chapter 1: Probability spaces, Discrete sample spaces (1.4 - 1.5), Conditional probabilities, Independent events (1.6 - 1.7)
3	Sep 9 - Sep 13	Chapter 1: The partition theorem (1.8), Probability measures, Worked problems, Problems (1.9 - 1.11), Chapter 2: Discrete random variables (2.1 - 2.2)
4	Sep 16 - Sep 20	Chapter 2: Functions of discrete random variables (2.3), Expectation, Conditional expectation (2.4 - 2.5), Problems (2.6)
5	Sep 23 - Sep 27	Chapter 3: Multivariate discrete distributions, Expectation (3.1 - 3.2), Independence of discrete random variables (3.3), Sums of random variables, Indicator functions (3.4 - 3.5)
6	Sep 30 - Oct 4	Chapter 3: Problems (3.6), Chapter 4: Probability generating functions (4.1 - 4.2)
7	Oct 7 - Oct 11	Chapter 4: Moments, Sums of independent random variables (4.3 - 4.4), Problems (4.5), Chapter 5: Distribution functions, Continuous random variables (5.1 - 5.2)
8	Oct 14 - Oct 18	Chapter 5: Examples of distribution functions, Continuous random variables (5.2 - 5.3), Some common density functions, Functions of random variables (5.4 - 5.5)
9	Oct 21 - Oct 25	Chapter 5: Expectations of continuous random variables (5.6), Geometrical probability, Problems (5.7 - 5.8)
10	Oct 28 - Nov 1	Chapter 6: Random vectors and independence (6.1 - 6.2), Joint density functions, Marginal density functions and independence (6.3 - 6.4)
11	Nov 4 - Nov 8	Chapter 6: Sums of continuous random variables, Changes of variables (6.5 - 6.6), Conditional density functions, Expectations of continuous random variables (6.7 - 6.8)
12	Nov 11 - Nov 15	Veterans Day (Nov 11, No Class), Chapter 6: Bivariate normal distribution, Problems (6.8 - 6.9), Chapter 7: Moments, Moment generating functions (7.1 - 7.3)
13	Nov 18 - Nov 22	Chapter 7: Variance and covariance, Moment generating functions (7.3 - 7.4), Two inequalities, Characteristic functions (7.5 - 7.6), Problems (7.7)
14	Nov 25 - Nov 29	Chapter 8: The law of averages, Chebyshev’s inequality (8.1 - 8.2), Thanksgiving Break (Nov 28 - Nov 29, No Class)
15	Dec 2 - Dec 6	Chapter 8: The Central Limit Theorem (8.3), Large deviations and Cramér’s theorem (8.4), Convergence in distribution, and characteristic functions (8.5), Problems (8.6)
16	Dec 9 - Dec 11	Review and Final Exam Preparation

Final Exam Week (Dec 13 - Dec 19)

Final Exam