Math 464, Theory of Probability

Syllabus for Math 464, Section 003, Fall 2021

Location and Times

This course meets TuTh 8-9:15am on Zoom, accessible to enrolled students via D2L.

Course Description

Course Prerequisites or Co-requisites

1. MATH 223 - Vector Calculus;
2. MATH 323 - Formal Mathematical Reasoning and Writing;
3. or equivalent coursework with instructor permission.

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.

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

Assignments and Examinations

Notes on exam administration

All examinations will be administered during the class time. They are to be held on Zoom in "gallery view" mode. The exam papers 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	October 5, 8-9:15am	20%
Midterm 2	November 16, 8-9:15am November 16-20 (due 11/20, 11:59pm; details in D2L)	20%
Final Examination	December 16, 8-10am	30%
Homework	See D2L	30%

Homework Assignments

Homework will consist of eight assignments, one per covered textbook chapter, and equally contributing to the grade, each worth 30/8 = 3.75% of the grade. The assignments are posted on line at this link: Homework. The assignment papers are collected via D2L.The due dates are posted in D2L only. Due dates will be adjusted as needed and the student is expected to check regularly for those adjustments.

Homework is assigned throughout the semester. Approximately 15 problems will be assigned per Chapter, for a total of 120 problems. Approximately 40% of each assignment will be considered extra credit, including the most difficult or time-consuming problems of each assignment. Routine problems will not be considered extra credit (e.g. exercises from the book which are not the problems from the last section of a Chapter; these last sections have problems of significant difficulty and many of them will be extra credit). Two types of homework will be assigned:

1. Homework which consists of selected exercises in the book.
2. Custom homework problems composed by the instructor.

Homework submission requirements

• Homework shall be submitted as a typed paper, with the exception of these graphs and figures which cannot be easily drawn with software. Recommended software for preparing the homework paper is \(\LaTeX\). Microsoft Word or similar software capable of formatting math formulas may also be used.
• Handwritten solutions will be not be accepted for full credit. However, If you have handwritten notes or untyped solutions, you should scan them and submit for partial credit for up to 75% of the maximum credit.
• The work shall be submitted electronically, as a PDF document, through D2L, using the Dropbox feature of D2L.
• A solution to each problem must start on a new page, with a boldface heading like Problem 1.11.3 (problem 3 in section 1.11).
• IMPORTANT: No e-mail submissions or paper submissions will be accepted.
• D2L distinguishes between 'Due Date' and 'End Date' for an assignment. Both dates are displayed in the Calendar tool in D2L. They will be further explained in class 'Announcements' in D2L. In general terms, the 'Due Date' is a 'soft deadline' after which the assignments still may be accepted; the 'End Date' is a 'hard deadline', after which submission will not be possible.

Homework preparation methods and software

The recommended method of homework preparation is using $\LaTeX$, although you may use other tools that are capable of producing high quality formulas. $\LaTeX$ is the tool of choice for writing math, CS and statistics papers, and therefore by using it you build skills which you can apply in the future.

Even if you choose $\LaTeX$, you still have many options in regard to software used in the document preparation. Here are some of my favorites:

• If you prefer to work on your desktop and have no experience with $\LaTeX$, I recommend TeXstudio. Note that old Mac OS users may require some work to get it working, but it is possible. Others should have no trouble installing.
• If you like working on-line, an excellent option is Overleaf with an interface very similar to TeXstudio, but running in Web browser. No installation is involved, and you will be able to work on your paper on multiple computers without copying files.
• If you are good with complex software and already know $\LaTeX$ above beginner level, you can useTeXmacs. This software has some remarkable capabilities, including running live "sessions" using several programming languages (including R, MATLAB, Maxima and Python). The software is WYSIWYG ("what you see is what you get") as opposed to $\LaTeX$ in which your homework is essentially a computer program. If you like simulations and experiments and you would like to include them in your homework solutions, TeXmacs may be for you. TeXmacs does not store documents in $\LaTeX$ format, but is capable of generating documents in $\LaTeX$. However, you would most likely generate PDF documents directly for submission to D2L.
• A computer algebra system (CAS) called Maxima allows you to calculate symbolically and generate formulas in $\TeX$ (the low-level typesetting system underneath $\LaTeX$). 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. Maxima is not able to generate full $\TeX$ documents, so you would use it to just typeset individual formulas. However, note that you can use Maxima from TeXmacs and in this manner create complete documents.

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)

      Date	Standard Class Dates: Fall 2021 - Undergraduate - Regular Academic Session
3/1/2021	Shopping Cart available
8/22/2021	Last day to file Undergraduate Leave of Absence
8/22/2021	Last day for students to add to or drop from a waitlist
8/23/2021	FIRST DAY OF FALL CLASSES

    UAccess still available for registration
    First day to file for the Grade Replacement Opportunity (GRO)
    First day to add classes for audit and instructor signature is required 

8/30/2021	Last day to use UAccess for adding classes, changing classes, or changing sections
8/31/2021	Instructor approval required on a Change of Schedule form to ADD or CHANGE classes
09/01/2021	Last day to apply for Fall degree candidacy without a late fee After this date a $50 00 Late Candidacy Application fee will be assessed
9/5/2021	

    Last day to drop without a grade of W (withdraw)
    Classes dropped on or before this date will remain on your UAccess academic record with a status of dropped, but will not appear on your transcript
    Last day to change from credit to audit, or vice versa, with only an instructor's signature

9/5/2021	Last day for a refund
9/6/2021	Beginning today, students may completely withdraw from all classes in the term
9/6/2021	Labor Day, no classes
9/6/2021	

    W period begins A penalty grade of W will be awarded for each withdrawal and the class(es) will appear on your transcript
    Beginning today, a change from credit to audit requires instructor approval on a Change of Schedule form

9/17/2021	Last day to change from pass/fail to regular grading or vice versa with only instructor approval on a Change of Schedule form
9/18/2021	Instructor's and dean's signatures are required on a Change of Schedule form to change from pass/fail to regular grades or vice versa
9/19/2021	Last day for department staff to add or drop in UAccess
10/17/2021	Last day to make registration changes without the dean's signature
10/18/2021	Instructor's and dean's signatures are required on all Change of Schedule forms to ADD or CHANGE classes
10/29/2021	Last day to file for Grade Replacement Opportunity (GRO)
10/31/2021	

    Last day for students to withdraw from a class online through UAccess
    Last day for students to change to/from audit with only instructor approval
    Last day for instructors to administratively drop students 

11/1/2021	

    Instructor and dean's signatures required on a Late Change Petition in order to withdraw from class and students must have an extraordinary reason for approval
    Instructor's and dean's permission required on a Change of Schedule form to change to/from audit

11/11/2021	Veteran's Day, no classes
11/21/2021	Last day for students to submit a Late Change Petition to their college
11/25/2021	Thanksgiving recess begins today with no classes until Monday
12/8/2021	Last day of class--no registration changes can be made after the last day of class
12/9/2021	Reading day, no classes
12/10/2021	Final exams begin
12/16/2021	

    Final exams end
    Final grades are available in UAccess as soon as the instructor posts them
    Per Faculty Senate Policy, grades should be submitted within two business days after the final exam 

12/17/2021	Degree award date

    

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
320	A		BASIC PROBABILITY
321	A	1	Events and probabilities	3
322	A	1.1	Experiments with chance	3
323	A	1.2	Outcomes and events	3
324	A	1.3	Probabilities	6
325	A	1.4	Probability spaces	7
326	A	1.5	Discrete sample spaces	9
327	A	1.6	Conditional probabilities	11
328	A	1.7	Independent events	12
329	A	1.8	The partition theorem	14
330	A	1.9	Probability measures are continuous	16
331	A	1.10	Worked problems	17
332	A	1.11	Problems	19
333	A	2	Discrete random variables	23
334	A	2.1	Probability mass functions	23
335	A	2.2	Examples	26
336	A	2.3	Functions of discrete random variables	29
337	A	2.4	Expectation	30
338	A	2.5	Conditional expectation and the partition theorem	33
339	A	2.6	Problems	35
340	A	3	Multivariate discrete distributions and independence	38
341	A	3.1	Bivariate discrete distributions	38
342	A	3.2	Expectation in the multivariate case	40
343	A	3.3	Independence of discrete random variables	41
344	A	3.4	Sums of random variables	44
345	A	3.5	Indicator functions	45
346	A	3.6	Problems	47
347	A	4	Probability generating functions	50
348	A	4.1	Generating functions	50
349	A	4.2	Integer-valued random variables	51
350	A	4.3	Moments	54
351	A	4.4	Sums of independent random variables	56
352	A	4.5	Problems	58
353	A	5	Distribution functions and density functions	61
354	A	5.1	Distribution functions	61
355	A	5.2	Examples of distribution functions	64
356	A	5.3	Continuous random variables	65
357	A	5.4	Some common density functions	68
358	A	5.5	Functions of random variables	71
359	A	5.6	Expectations of continuous random variables	73
360	A	5.7	Geometrical probability	76
361	A	5.8	Problems	79
362	B		FURTHER PROBABILITY
363	B	6	Multivariate distributions and independence	83
364	B	6.1	Random vectors and independence	83
365	B	6.2	Joint density functions	85
366	B	6.3	Marginal density functions and independence	88
367	B	6.4	Sums of continuous random variables	91
368	B	6.5	Changes of variables	93
369	B	6.6	Conditional density functions	95
370	B	6.7	Expectations of continuous random variables	97
371	B	6.8	Bivariate normal distribution	100
372	B	6.9	Problems	102
373	B	7	Moments, and moment generating functions	108
374	B	7.1	A general note	108
375	B	7.2	Moments	111
376	B	7.3	Variance and covariance	113
377	B	7.4	Moment generating functions	117
378	B	7.5	Two inequalities	121
379	B	7.6	Characteristic functions	125
380	B	7.7	Problems	129
381	B	8	The main limit theorems	134
382	B	8.1	The law of averages	134
383	B	8.2	Chebyshev’s inequality and the weak law	136
384	B	8.3	The central limit theorem	139
385	B	8.4	Large deviations and Cramér’s theorem	42
386	B	8.5	Convergence in distribution, and characteristic functions	145
387	B	8.6	Problems	149
388	C		RANDOM PROCESSES
389	C	9	Branching processes	157
390	C	9.1	Random processes	157
391	C	9.2	A model for population growth	158
392	C	9.3	The generating-function method	159
393	C	9.4	An example	161
394	C	9.5	The probability of extinction	163
395	C	9.6	Problems	165
396	C	10	Random walks	167
397	C	10.1	One-dimensional random walks	167
398	C	10.2	Transition probabilities	168
399	C	10.3	Recurrence and transience of random walks	170
400	C	10.4	The Gambler’s Ruin Problem	173
401	C	10.5	Problems	177
402	C	11	Random processes in continuous time	181
403	C	11.1	Life at a telephone switchboard	181
404	C	11.2	Poisson processes	183
405	C	11.3	Inter-arrival times and the exponential distribution	187
406	C	11.4	Population growth, and the simple birth process	189
407	C	11.5	Birth and death processes	193
408	C	11.6	A simple queueing model	195
409	C	11.7	Problems	200
410	C	12	Markov chains	205
411	C	12.1	The Markov property	205
412	C	12.2	Transition probabilities	208
413	C	12.3	Class structure	212
414	C	12.4	Recurrence and transience	214
415	C	12.5	Random walks in one, two, and three dimensions	217
416	C	12.6	Hitting times and hitting probabilities	221
417	C	12.7	Stopping times and the strong Markov property	224
418	C	12.8	Classification of states	227
419	C	12.9	Invariant distributions	231
420	C	12.10	Convergence to equilibrium	235
421	C	12.11	Time reversal	240
422	C	12.12	Random walk on a graph	244
423	C	12.13	Problems	246
424	Appendix A		Elements of combinatorics	250
425	Appendix B		Difference equations	252