Math 421-01,02 (Fall 2019)
Probability Theory, Fall 2019, Math 421 - 01 & 02
You will want to check back here regularly as I may post a variety of course materials or links to various resources.
Syllabus
Homework:
HW 1: Ch. 1 # 8,9,10,14,27,31,37,41,42,52,54 (due: F 9/6) answer key (note this is just answers with minimal explanation)
HW 2: Ch. 2 # 7,8,9,15,17,39aegi,43,50,54,55,60,63 (due: F 9/13) answer key
HW 3: Ch. 2 # 21,22,24,76,94,108
Ch. 3 # 2,4,7,13,20,24,34 (due: F 9/20) answer key (except #7)
HW 4: Ch. 3 # 42,43,51,53,56,57,68,69,71,74,77,109 (due: F 9/27)
*note the solution in the book for # 57 is incorrect!
The correct answer is 4*exp(-3).
Hint: split the region of integration into two pieces. answer key & soln for 3.57 *(fixed)
HW 5: Ch. 4 # 5,6,7,8,12,13,14,19,22,23,24,29,31,33,34,37 (due: F 10/11) answer key
*note 4.33 has a typo in the answer in the book.
The correct answer is M(t)=(e^t-1)/t
HW 6: Ch. 4 # 40,44,45,47,49,50,51,56,58,60 (due: F 10/18) answer key
HW 7: Ch. 5 # 1,22,23,24,41,50,51,60,62,81 (due: M 11/4) answer key
HW 8: Ch. 6 # 1,3,15,16,21,22,23,24,36,37,38,41,42,47,54,59,61,63,67,80 (due: M 11/11) answer key
HW 9: Ch. 7 # 1,2,3,11,17,43,44,45,52,
Ch. 8 # 1,2,3,4,18,20,21,22,65,71,77,80,81,82 (due: F 12/6)
Other Assignments:
Quiz 1 (take home, due M 9/23)
Quiz 2 (take home, due M 11/11) solutions
Exam 1 bonus preview
Exam 2 bonus preview
Worksheet on discrete distributions
Final Exam part I (take home, due by Thurs 12/12 8am)
Review materials:
Here I will post additional review materials such as formula sheets or problems/solutions.
Chapter 1 review
Chapter 2 review
Chapter 3 review
For Chapters 1-3 review, you can also see my Math 321 Chapter 2 review here.
Chapter 4 review (*typos fixed again, E(X^2)=4)
Chapter 5 review
Chapter 6 review (*typos fixed, and gamma RV text fixed, normal mgf variance typo fixed)
Bivariate Normal Desmos contour plot
Exam 2 formula sheet (*updated again, mgf for normal fixed)
Final Exam formula sheet
Chapter 7 review
Chapter 8 review (The answer to 1a is 0.15%, i.e. (1-0.997)/2, typo fixed.)
Course notes:
You are advised to acquire a copy of the textbook. I will provide some of my own notes as well in order to supplement the text.
You are free to look at my Summer 2019 Math 321 course page where I have the most recent version of my own probability/statistics notes. These notes are not comprehensive over what Math 421 covers, but you might find them helpful. Also, there are many review problems with solutions on that course page as well. Chapters 2,3, and 4 are most relevant for Math 421. Chapters 5,6,7 and 8 will be relevant for Math 422. Math 421 and 422 together will cover the all of the material from Math 321 but at a greater theoretical depth and a greater number of topics.
R statistical software:
You are advised to gain access to R statistical software. Here are instructions for getting R statistical software up and running on your computer (both for MAC and Windows). It should already be installed in all computer labs in Herak. If you find a computer where it is not installed, you can use these instructions to install it, or report it to me, and I will have IT install it.
Another option for having quick online access to R (and this is useful for a smartphone) is rdrr: https://rdrr.io/snippets/. This is just limited to computations not exceeding 10 seconds or so.
SageCell at: https://sagecell.sagemath.org/ is another useful option. This website can be used to evaluate commands from a variety of programming languages (including MATLAB and Python). Just select R from the language tab at the lower right of the textbox. If you are familiar with MATLAB, choose the option "Octave". Octave is essentially an open source version of MATLAB and you can run MATLAB code using the Octave language option on SageCell.
R miscellaneous:
Here is a website where you can evaluate R code online from your web browser from any device: https://rdrr.io/snippets/
Coin flipping simulation code.
Poker simulation code.
Course plan:
Here is a weekly outline of the course and material covered. This is a tentative plan, but we will likely not vary from this by more than a few days.
| Week | Dates | Topic | Holidays | Chapters |
| 1 | 8/26 to 8/30 | Combinatorics | no class M | 1 |
| 2 | 9/2 to 9/6 | Probability | no class M | 2 |
| 3 | 9/9 to 9/13 | Probability | 2 | |
| 4 | 9/16 to 9/20 | Random Variables | 2,3 | |
| 5 | 9/23 to 9/27 | Distributions | 3 | |
| 6 | 9/30 to 10/4 | Joint Distributions | 3 | |
| 7 | 10/7 to 10/11 | Expectation | 3,4 | |
| 8 | 10/14 to 10/18 | Discrete Distr. | 4,5 | |
| 9 | 10/21 to 10/25 | Discrete Distr. | no class M | 5,6 |
| 10 | 10/28 to 11/1 | Cont. Distr. | 6 | |
| 11 | 11/4 to 11/8 | Cont. Distr. | 6,8 | |
| 12 | 11/11 to 11/15 | Sampling Distr. | 6,7 | |
| 13 | 11/18 to 11/22 | Sampling Distr. | 7,8 | |
| 14 | 11/25 to 11/29 | Functions of RVs | no class WRF | 7,8 |
| 15 | 12/2 to 12/6 | Functions of RVs | 7,8 | |
| 16 | 12/9 to 12/13 | Final Exam |
Tentative exam dates:
Here are tentative exam dates. These dates may change, but probably not by more than +/- a few days.
Exam 1, Monday 9/30, Chapters 1, 2, & 3
Exam 2, Monday 11/11, Chapters 4, 5, & 6
The final exam date and time is set by the university as follows:
Final Exam, comprehensive over entire course, Chapters 1-8
Section 2, MWF 10:00am Final Exam date/time: Tuesday, December 10 1:00 p.m. - 3:00 p.m.
Section 1, MWF 11:00am Final Exam date/time: Thursday, December 12 8:00 a.m. - 10:00 a.m.
Studying and learning advice:
Studying is generally a very personal endeavor. You are advised to figure out what works for you. For most, mathematics is best learned by active engagement. Solving many problems and performing many calculations is advised. Often the best learning occurs when a problem is attempted multiple times with failure before succeeding.
Reflection is a key component to learning. Whether you solved a problem correctly or not or if you do or don't feel like you understand a particular concept, focusing on it with intent and active attention will help you make progress. Solving the same problem multiple times even if you already achieved the correct solution will help hard-wire the concepts and methodology in your brain. Try different approaches. Check for similar examples online or in other course materials. Mathematics Stack Exchange has a great wealth of information as well where many students have probably already asked the same questions you have. There are many other online math forums as well.
Persistence is absolutely necessary! You will be rewarded by your efforts!
A note about the quizzes, exams, and material coverage:
Quizzes generally only cover the material learned in the previous week or two. Each quiz is worth 25 points, and this can be thought of symbolizing that it is roughly one quarter of a regular exam in length. Normally quizzes are from 1-2 pages in length. They are timed for 15 to 25 minutes. The actual content may vary from 1-2 longer problems (i.e. longer more involved calculations) or 6-8 short problems (e.g. quick calculations, short answer, checking basic concepts, terminology, or multiple choice).
Regular exams only cover the new material since the last regular exam. Each regular exam is worth 100 points. Regular exams are normally 4-6 pages long and are timed for 50 minutes. You can use the quizzes as a rough guide for the exams, i.e. a regular exam will be roughly equivalent to four quizzes.
The Final Exam will cover any new material since the last regular exam and/or quiz and additionally will be comprehensive over the entire course. It is timed to 120 minutes, and thus can hypothetically be 2.4x the length of a regular exam. As a general guideline, you can expect it to be roughly equivalent to two complete exams.
These are only general guidelines and are meant to help you prepare for the quizzes and exams. Exams and quizzes are constructed so as to be able to be completed in the allotted time for a student who has sufficient preparation through attending class, studying, and completing all assignments with intention and effort.
Studying and learning advice:
Studying is generally a very personal endeavor. You are advised to figure out what works for you. For most, mathematics is best learned by active engagement. Solving many problems and performing many calculations is advised. Often the best learning occurs when a problem is attempted multiple times with failure before succeeding.
Reflection is a key component to learning. Whether you solved a problem correctly or not or if you do or don't feel like you understand a particular concept, focusing on it with intent and active attention will help you make progress. Solving the same problem multiple times even if you already achieved the correct solution will help hard-wire the concepts and methodology in your brain. Try different approaches. Check for similar examples online or in other course materials. Mathematics Stack Exchange has a great wealth of information as well where many students have probably already asked the same questions you have. There are many other online math forums as well.
Persistence is absolutely necessary! You will be rewarded by your efforts!
A note about the quizzes, exams, and material coverage:
Quizzes generally only cover the material learned in the previous week or two. Each quiz is worth 25 points, and this can be thought of symbolizing that it is roughly one quarter of a regular exam in length. Normally quizzes are from 1-2 pages in length. They are timed for 15 to 25 minutes. The actual content may vary from 1-2 longer problems (i.e. longer more involved calculations) or 6-8 short problems (e.g. quick calculations, short answer, checking basic concepts, terminology, or multiple choice).
Regular exams only cover the new material since the last regular exam. Each regular exam is worth 100 points. Regular exams are normally 4-6 pages long and are timed for 50 minutes. You can use the quizzes as a rough guide for the exams, i.e. a regular exam will be roughly equivalent to four quizzes.
The Final Exam will cover any new material since the last regular exam and/or quiz and additionally will be comprehensive over the entire course. It is timed to 120 minutes, and thus can hypothetically be 2.4x the length of a regular exam. As a general guideline, you can expect it to be roughly equivalent to two complete exams.
These are only general guidelines and are meant to help you prepare for the quizzes and exams. Exams and quizzes are constructed so as to be able to be completed in the allotted time for a student who has sufficient preparation through attending class, studying, and completing all assignments with intention and effort.