M 362K: Probability I (Spring 2012)
This is the course page for James Pascaleff's M 362K, unique number 55915.
Table of Contents
Final Exam Announcements
- Final Exam
- Friday, May 11, 2:00-5:00 pm, in RLM 6.122
- Final Exam Office Hours
- Monday May 7, 1-4pm and Wednesday May 9, 9:30am-12pm
- Note Sheets
- Two sheets of notes, front and back (= 4 sides total), are permitted at the final exam.
- Study Aids
-
All lecture notes in one PDF (26 MB, 258 pp.): All Lectures,
All homework solutions in one PDF (19 MB, 130 pp.): All HW Solutions. - Solutions
- Here is the final exam: Exam 4 (w/o solutions), and Exam 4 solutions.
Vital information
- Lecture
- MWF 11:00-12:00 noon in RLM 6.122
- Instructor: James Pascaleff
- Email: jpascaleff@math.utexas.edu Office: RLM 11.166 Office hours: M 1:00-2:00 pm, 3:00-4:00 pm, W 9:30-10:30 am
- Textbook
- A First Course in Probability by Sheldon Ross. 8th Edition.
- Prerequisite
- M 408D, 408L, or 408S with a grade of at least C-.
- Homework
- Starting Wednesday, January 25, homework assignments are due on Wednesdays in class.
- Midterm Exams
- Friday, February 17 Friday, March 23 Friday, April 20
- Final Exam
- Friday, May 11, 2:00-5:00 pm, in RLM 6.122
- Grading
-
Homework 18% Midterm Exam 1 18% Midterm Exam 2 18% Midterm Exam 3 18% Final Exam 28% - Grade Scale
-
A 93-100 C 73-76 A- 90-92 C- 70-72 B+ 87-89 D+ 67-69 B 83-86 D 63-66 B- 80-82 D- 60-62 C+ 77-79 F 0-59
Course Description
This is an introductory course in the mathematical theory of probability. It is fundamental to further work in probability and statistics.
The first part of the course introduces set theory and a set of axioms for probability, using them to understand basic probability properties, conditional probability, and independence. The concept of a random variable, expectation, and variance are also introduced. We will develop some combinatorial techniques for solving problems in discrete probability, and we will study some common distributions associated with random variables.
Starting with the material in Chapter 5, we will develop the theory of continuous random variables using techniques from calculus (including some ideas from multivariable calculus). Jointly distrubuted random variables and the properties of expectation play a key role in the second half of the course, whose main goal is the celebrated Central Limit Theorem and the Law of Large Numbers.
Course Plan & Lecture Notes
This course plan will be fleshed out as the semester progresses. It shows the approximate dates when we will begin each chapter in the text. Some of the days are marked "flexible" which means that they may end up being used for review, extra topics, moving ahead, or catching up if we get behind schedule.
| Date | Topic | Lecture Notes |
| W 1/18 | Basic principle of counting, permutations | Lecture 1 |
| F 1/20 | Indistinguishable objects, combinations | Lecture 2 |
| M 1/23 | The Binomial Theorem | Lecture 3 |
| W 1/25 | Sample spaces and events | Lecture 4 |
| F 1/27 | Axioms of probability | Lecture 5 |
| M 1/30 | Basic properties, inclusion-exclusion | Lecture 6 |
| W 2/1 | Equally likely outcomes | Lecture 7 |
| F 2/3 | Equally likely outcomes, cont. | Lecture 8 |
| M 2/6 | Conditional probabilities | Lecture 9 |
| W 2/8 | Bayes's formula | Lecture 10 |
| F 2/10 | Independent events | Lecture 11 |
| M 2/13 | Conditional probability as a probability | Lecture 12 |
| W 2/15 | Review | Lecture 13 |
| F 2/17 | Midterm Exam 1 | |
| M 2/20 | Random variables | Lecture 14 |
| W 2/22 | Discrete RVs, Probabilty mass function | Lecture 15 |
| F 2/24 | Expectation | Lecture 16 |
| M 2/27 | Properties of Expectation | Lecture 17 |
| W 2/29 | Variance, binomial random variables | Lecture 18 |
| F 3/2 | Binomial and Poisson random variables | Lecture 19 |
| M 3/5 | Bernoulli, Poisson and random processes | Lecture 20 |
| W 3/7 | Other discrete random variables | Lecture 21 |
| F 3/9 | Continuous random variables | Lecture 22 |
| M 3/12 | Spring Break | |
| W 3/14 | Spring Break | |
| F 3/16 | Spring Break | |
| M 3/19 | Continuous RVs, expectation and variance | Lecture 23 |
| W 3/21 | Review | Lecture 24 |
| F 3/23 | Midterm Exam 2 | |
| M 3/26 | Example of expectation | Lecture 25 |
| W 3/28 | Properties of expectation & uniform random variables | Lecture 26 |
| F 3/30 | Uniform and Normal random variables | Lecture 27 |
| M 4/2 | Normal RVs and DeMoivre-Laplace limit theorem | Lecture 28 |
| W 4/4 | Exponential and Gamma random variables | Lecture 29 |
| F 4/6 | Functions of random var's and Joint distributions | Lecture 30 |
| M 4/9 | Joint distributions and independence | Lecture 31 |
| W 4/11 | Sums of independent variables | Lecture 32 |
| F 4/13 | Conditional distributions | Lecture 33 |
| M 4/16 | Expectation of sums of random variables | Lecture 34 |
| W 4/18 | Review | Lecture 35 |
| F 4/20 | Midterm Exam 3 | |
| M 4/23 | Covariance and correlation | Lecture 36 |
| W 4/25 | Chebyshev's inequality and Weak Law of Large Numbers | Lecture 37 |
| F 4/27 | The Central Limit Theorem | Lecture 38 |
| M 4/30 | The Strong Law of Large Numbers | Lecture 39 |
| W 5/2 | Proof of the central limit theorem | Lecture 40 |
| F 5/4 | Final review | Lecture 41 |
| All Lecture notes in one PDF (26 MB, 258 pp.) | All Lectures |
Homework
Written homework is due in class on the due date.
| Due Date | Problems | Solutions | Remarks |
|---|---|---|---|
| W 1/25 | pp. 16-17: 1,3,5,8,9,11,13,28 | Solutions | |
| pp. 18-19: 2,3,4,8,10 | |||
| W 2/1 | pp. 18-19: 13,14 | Solutions | |
| p. 50: 1,2,3,4,5 | |||
| pp. 54-55: 1,2,3,6,7,9 | |||
| W 2/8 | pp. 51-54: 8,13,18,22,25,32,39,41,54 | Solutions | |
| p. 55: 10,11,18,19 | |||
| p. 110: 3.2 | |||
| W 2/15 | pp. 102-109: 3.12,3.22,3.24,3.37,3.47,3.66,3.78 | Solutions | |
| pp. 110-112: 3.4,3.6,3.10,3.16,3.18 | |||
| W 2/22 | p. 113: 3.26,3.28,3.29 | Solutions | |
| pp. 172-173: 4.1,4.3,4.5,4.6 | |||
| W 2/29 | pp. 173-175: 4.13,4.18,4.19,4.20,4.28 | Solutions | |
| p. 180: 4.2,4.3,4.4,4.7,4.8 | |||
| W 3/7 | pp. 175-177: 4.30,4.33,4.38,4.41,4.57,4.59,4.61 | Solutions | p. 176, problem 4.35 is an extra credit problem |
| pp. 180-181: 4.16,4.19 | (It was assigned in the notes, but dropped from this page.) | ||
| W 3/21 | pp. 177-179: 4.63,4.70,4.71,4.75,4.78 | Solutions | |
| p. 182: 4.27,4.29 | |||
| p. 224: 5.1 | |||
| W 3/28 | pp. 224-225: 5.2,5.3,5.4,5.6,5.7,5.8 | Solutions | |
| W 4/4 | pp. 225-226: 5.12,5.13,5.15,5.16,5.18,5.23,5.27 | Solutions | |
| pp. 227-228: 5.2,5.3,5.9 | |||
| W 4/11 | pp. 226-227: 5.32,5.34,5.39 | Solutions | |
| pp. 228-229: 5.13,5.30,5.31 | |||
| p. 287: 6.1,6.7,6.9 | |||
| W 4/18 | pp. 228-290: 6.18,6.20,6.29,6.30,6.32,6.38,6.42 | Solutions | |
| pp. 291-292: 6.11,6.14 | |||
| W 4/25 | p. 373: 7.5,7.9,7.11 | Solutions | |
| p. 380: 7.4,7.5 | |||
| W 5/2 | p. 375: 7.32,7.36,7.38 | Solutions | |
| p. 381: 7.19 | |||
| pp. 412-413: 8.1,8.4,8.5,8.7 | |||
| p. 414: 8.1 | |||
| All homework solutions in one PDF (19 MB, 130 pp.) | All Solutions |
Homework policies
Starting January 25, homework will be due each Wednesday in class. The homework will consist of written exercises. This website is the definitive source for the homework assignments.
Over the course of the semester, there will 14 homework assignments. The two lowest homework scores are dropped when computing the homework score. The homework counts for 18% of the course grade, so each of the 12 highest homework scores counts for 1.5% of the course grade.
Late homework will not be accepted. Because the two lowest scores are dropped, you can miss one or two assignments without penalty.
Many of the problems on the written homework will require you not only to provide an answer but to justify it. This means that you must show the steps that lead to your answer, and that the correct answer by itself will not necessarily yield full credit.
You may work with other students on written homework, but you must write up your solutions by yourself. Please indicate on your homework the students that you work with.
The grader is only paid for a certain number of hours per week, so depending on the length of the homework assignments, it may not be possible for the grader to grade every homework problem in detail. If this is the case, the instructor will select some problems for the grader to grade in detail, while grading the others for completion.
Exams
The three midterm exams will be given in class, at the usual time and location. Each midterm exam is worth 18% of the final grade. The exams are cumulative, but each focuses on one portion of the course. The dates and topics of the exam are:
| Date | Topics | Exam | Solutions | |
| Midterm Exam 1 | Friday, February 17 | Chapters 1, 2, and 3 (through Lecture 11) | Exam 1 | Solutions |
| Midterm Exam 2 | Friday, March 23 | Chapter 4, beginning of 5 (through Lecture 22) | Exam 2 | Solutions |
| Midterm Exam 3 | Friday, April 20 | Chapters 5 and 6. | Exam 3 | Solutions |
| Final Exam | Friday, May 11 | Comprehensive | Exam 4 | Solutions |
The final exam covers the entire course, with possibly some focus on Chapters 7 and 8 of the text (since these sections are not covered by the other exams).
| Final Exam | Friday, May 11, 2:00-5:00 pm | Location: RLM 6.122 |
Exam policies
No books or calculators are allowed on exams.
Starting with the second midterm exam, a sheet of notes is allowed. It must conform to the following standards:
- 1 US Letter (8.5 inch by 11 inch) sheet of paper, both sides of which may be used.
- It must be handwritten by the student. No Xeroxing is allowed.
- You may put as much as you want in the space available, but no magnifying glasses may be used during the exam.
- For the final exam, 2 sheets of notes are permitted.
Students who have prior commitments that interfere with the exam times should let the instructor know before the 12th class day, February 1st, so that appropriate accomodations can be made.
If you are unable to take an exam due to an emergency, you must let the instructor know as soon as you yourself become aware that you are likely to miss the exam. It is particularly important that you notify the instructor before the beginning of the exam. Otherwise, you will recieve a grade of 0 on the exam. Make-up exams or other accomodations will be offered for excused absences as appropriate to each student's situation.
The final exam will be offered only at the time set by the Registrar. Extraordinary circumstances that cause a student to miss the final exam will be handled in accordance with the policies of the College of Natural Sciences and the University.
Grading
Homework contributes 18% to the final grade. Each of the three midterm exams contributes 18%, and the final exam contributes 28%. Attendance does not contribute directly to the final grade.
Grade Scale
| A | 93-100 | C | 73-76 |
| A- | 90-92 | C- | 70-72 |
| B+ | 87-89 | D+ | 67-69 |
| B | 83-86 | D | 63-66 |
| B- | 80-82 | D- | 60-62 |
| C+ | 77-79 | F | 0-59 |
Deadlines for dropping the course
- 12th class day: Wednesday, February 1
- Q-drop deadline: Monday, April 2
- If you drop a class on or before February 1, the class will not show up on your transcript. If you drop a class after that date, the course will show up on the transcript with a grade of "Q". After April 2, it is not possible to drop a course except for extenuating (usually non-academic) circumstances.
Religious Holidays
In accordance with UT Austin policy, please notify the instructor at least 14 days prior to the date of observance of a religious holiday. If you cannot complete a homework assignment in order to observe a religious holiday, you will be excused from the assignment. If the holiday conflicts with an exam, you will be allowed to write a make-up exam within a reasonable time.
Special Needs
Students with disabilities may request appropriate academic accommodations from the Division of Diversity and Community Engagement, Services for Students with Disabilities at 471-6259 (voice), 232-2937 (video), or http://www.utexas.edu/diversity/ddce/ssd.
Academic Integrity
If you work with other students, indicate this on your homework. You must write up your homework by yourself. Read the University's standard on academic integrity found on the Student Judicial Services website.