Math/Stat 431 Introduction to the Theory of Probability - Lecture 4

Fall 2016, Lecture 4

Meetings: MWF, 2:25-3:15 PM Van Vleck B119

Instructor: Daniele Cappelletti

Office: 415 Van Vleck.

Office Hours: Wednesday 3.30 - 5.30pm, and by appointment.

E-mail:cappelletti@math.wisc.edu

This is the course homepage that also serves as the syllabus for the course. Here you will find our weekly schedule, and updates on scheduling matters. The Mathematics Department also has a general information page on this course.

I will use the class email list to send out corrections, announcements, etc. Please check your wisc.edu email regularly.

Course description

Math 431 provides an introduction to the theory of probability, the part of mathematics that studies random phenomena. We model simple random experiments mathematically and learn techniques for studying these models. Topics covered include the axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

431 is not a course in statistics. Statistics is a discipline mainly concerned with analyzing and representing data. Probability theory forms the mathematical foundation of statistics, but the two disciplines are separate.

From a broad intellectual perspective, probability is one of the core areas of mathematics with its own distinct style of reasoning. Among the other core areas are analysis, algebra, geometry/topology, logic and numerical analysis.

To go beyond 431 in probability you should take next 521 -- Analysis, and after that one or both of these: 632 Introduction to Stochastic Processes and 635 Introduction to Brownian Motion and Stochastic Calculus. Those who would like a proof based introduction to probability could consider taking Math 531 - Probability Theory (531 requires a proof based course as a prerequisite).

Where is probability used?

Probability theory is used ubiquitously throughout the sciences and industry. For example, in biology many models of cellular phenomena are now modeled probabilistically as opposed to deterministically (this is the subject area that got me interested in probability). As for industry, many models used by insurance companies are probabilistic in nature (see: actuarial science). Also, many of the models used by the finance industry are also probabilistic in nature (google "Black-Scholes" to see an example). Thus, those wishing to go into finance need to have a solid understanding of probability.

Prerequisites

To be technically prepared for Math 431 one needs to be comfortable with the language of sets and calculus, including multivariable calculus, and be ready for abstract reasoning. Basic techniques of counting is also useful, but we will review these along the way. Probability theory can seem very hard in the beginning, even after success in past math courses. It is imperative that you be willing to put in the required effort.

Textbook

The course follows lecture notes by David Anderson, Timo Sepp\E4l\E4inen, and Benedek Valk\F3. These will be provided to the students at no cost. The textbook can be found at the Learn@UW website. The following textbooks can be used as a resource for extra practice problems. They have been placed on reserve at the math library (floor B2 of Van Vleck).

A first course in Probability, by Sheldon Ross
Probability, by Jim Pitman.
Probability, statistics, and stochastic processes, by Peter Olofsson.
Probability and random processes, by Geoffrey Grimmett and David Stirzaker.

Learn@UW

We will use the Learn@UW website of the course to post homework assignments and solutions. The lecture notes will also be posted there.

Piazza
We will be using Piazza for class discussion. The system is catered to getting you help fast and efficiently from classmates and myself. Rather than emailing math questions to me, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com.

Find our class Piazza page here.

Socrative
During the lectures we will make use of the software Socrative, which can be used for free. The software provides a service similar to that provided by clickers: I ask you questions and you can answer anonymously. To be able to use it, you need to bring your smartphone or your laptop to lectures, visit the webpage
www.socrative.com
and login as a student. When you are asked for a name, type "MATH431" (without quotes). Alternatively, you can download the student app here, and enter "MATH431" as room name.

Evaluation

Course grades will be based on homework and quizzes (15%), two midterm exams (2x25%), and a comprehensive final exam (35%). Your lowest quiz grade and your lowest homework grade will be dropped. As part of your homework, you are asked to use Webwork, an online platform where you can enter the solution of some exercises (Scroll down for more information).

Midterm exam 1: Thursday, October 13th, 7:15 PM - 8:45 PM, 1310 Sterling.
Make-up exam for Midterm exam 1: Friday, October 14, 7:15 AM - 8:45 AM, B115 Van Vleck. Please inform me as soon as possible if you intend to take the make-up instead of the regular exam. Note that the make-up exam is in the morning!
Midterm exam 2: Wednesday, November 30th, 7:15 PM - 8:45 PM, 1310 Sterling.
Make-up exam for Midterm exam 1: Thursday, December 1, 8:45 AM - 10:15 AM, 1323 Sterling. Please inform me as soon as possible if you intend to take the make-up instead of the regular exam. Note that the make-up exam is in the morning!
Final exam: Tuesday, December 20th, 12:25 PM - 2:25 PM, SOC SCI 5231.

No calculators, cell phones, or other gadgets will be permitted in exams and quizzes, only pencils, pens and paper.

The final grades will be determined according to the following scale:
A: [100,89), AB: [89,87), B: [87,76), BC: [76,74), C: [74,62), D: [62,50), F: [50,0].
There will be no curving in the class, but the instructor reserves the right to modify the final grade lines.

Bonus problems will be provided on the homework assignments. Successful completion of these more challenging problems can lead to a few extra percentage points being added to your final grade.

Quizzes

To help prepare for the midterm exams we will have short in-class quizzes during the first few weeks. You will be able to find the quiz dates below.

First quiz: September 12, Monday, end of class.
Second quiz: September 19, Monday, end of class.
Third quiz: September 26, Monday, end of class.
Fourth quiz: October 3, Monday, end of class.

The purpose of the quizzes is to give you practice answering probabilistic questions in a timed environment. We will have these quizzes until I decide they are no longer needed. No makeup quizzes will be given, instead you may drop your lowest quiz score.

Homework

Homework assignments will be posted on the Learn@UW site of the course. Weekly homework assignments are due Fridays at the beginning of the class. Note that there is a (short) homework assignment due on the first Friday (September 9).

Instructions for homework

Observe rules of academic integrity. Handing in plagiarized work, whether copied from a fellow student or off the web, is not acceptable. Plagiarism cases will lead to sanctions.
Homework is collected at the beginning of the class period on the due date. No late papers will be accepted. You can bring the homework earlier to the instructor's office or mailbox.
Working in groups on homework assignments is strongly encouraged; however, every student must write their own assignments.
Organize your work neatly. Use proper English. Write in complete English or mathematical sentences. Answers should be simplified as much as possible. If the answer is a simple fraction or expression, a decimal answer from a calculator is not necessary. But for some exercises you need a calculator to get the final answer.

Answers to some exercises are in the back of the book, so answers alone carry no credit. It's all in the reasoning you write down.
Put problems in the correct order and staple your pages together.
Do not use paper torn out of a binder.
Be neat. There should not be text crossed out.
Recopy your problems. Do not hand in your rough draft or first attempt.
Papers that are messy, disorganized or unreadable cannot be graded.
I strongly encourage you to type up your solutions (perhaps using Latex).

Succeeding in this course: This course will be more difficult than previous math courses, although I strongly believe it will be more rewarding for those that put in the necessary effort. To succeed you should read the sections of the text before each lecture and then again after. You should work together in small groups and discuss the material. (You may use Piazza to put a group together.) Such discussions are an invaluable way to learn the material. After working in the groups, you should carefully and clearly write-up your solutions making sure you actually understand the material. If, even after completing a homework assignment, you feel you still do not fully understand the material as well as you could (perhaps because you could not follow the discussion in your working group well), you should do more problems.

Webwork

During the course you will be asked to do some exercises on an online platform called Webwork. You can find it here: http://webwork.math.wisc.edu/webwork2
You should have been provided with credential to access "math431_fall2016", it may be a good idea to change your password once you have logged in.
To enter the solution, you just need to type it in. Webwork interface accepts numbers, fractions, functions, etc. , with the syntax that you would expect. While you do not have the possibility to explain your argument on Webwork, you can check whether your solution is correct, so you have an immediate feedback on whether you are on the right path. Also, if your Webwork solution is wrong, you can try again as many times as you like, until the due date. This can of course be helpful to you, as you may be able to assess what you understood on your own, and in case correct yourself before writing down the solution of the homeworks on paper.
The Webwork exercises as a whole (all the online exercises) will have the same weight as a single 'regular' homework towards the final grade.

Weekly schedule

Here is a tentative weekly schedule, to be adjusted as we go. The numbers refer to sections in lecture notes that can be found at the Learn@UW website.

Week	Monday	Wednesday	Friday
1 9/6-9/9		Appendix A Set notation. 1.1 Sample spaces and probabilities.	HW 1 due. 1.2 Equally likely outcomes.
2 9/12-9/16	Quiz #1. 1.2 Equally likely outcomes. 1.3 Infinitely many outcomes.	1.3 Infinitely many outcomes. 1.4 Consequences of the rules of probability.	HW 2 due. 1.4 Consequences of the rules of probability. 1.5 Random variables:a first look.
3 9/19-9/23	Quiz #2. 1.5 Random variables:a first look.	2.1 Conditional probability.	HW 3 due. 2.2 Bayes' formula 2.3 Independence.
4 9/26-9/30	Quiz #3. 2.3 Independence	2.3 Independence 2.4 Independent trials	HW 4 due. 2.4 Independent trials from 2.5: "Conditional independence" and "The birthday problem"
5 10/3-10/7	Quiz #4. 3.1 Distributions of random variables	3.2 Expectations and variance.	HW 5 due. 3.2 Expectations and variance. 3.3 The Gaussian distribution.
6 10/10-10/14	3.3 The Gaussian distribution.	Exam 1 in evening (moved to Thursday). 3.3 The Gaussian distribution.	3.3 The Gaussian distribution.
7 10/17-10/21	3.3 The Gaussian distribution. 4.1 Normal approximation: The CLT for binomial and examples	4.1 Normal approximation: continuity correction, law of large numbers and confidence intervals.	HW 6 due. 4.1 Normal approximation: confidence intervals.
8 10/24-28	4.2 Poisson approximation	4.2 Poisson approximation 4.3 Exponential distribution	HW 7 due. 4.3 Exponential distribution 5.1 Moment generating functions (skipped 4.4 The Poisson Process)
9 10/31-11/4	5.1 Moment generating functions	5.2 Distribution of a function of a random variable	HW 8 due. 6.1 Joint distribution of discrete random variables. 6.3 Cumulative distribution function and expectations for several random variables (only Fact 6.41 and Example 6.42)
10 11/7-11/11	6.1 Joint distribution of discrete random variables.	6.2 Jointly continuous random variables 6.3 Cumulative distribution function and expectations for several random variables (only Fact 6.41)	HW 9 due. 6.2 Jointly continuous random variables 6.3 Cumulative distribution function and expectations for several random variables (only Example 6.44)
11 11/14-11/18	7.1 Sums of independent random variables.	7.1 Sums of independent random variables.	7.2 Exchangeable random variables. (skipped 7.3 The Poisson Process) 8.1 Linearity of expectation.
12 11/21-11/25	HW 10 due. 8.2 Expectation and independence	Coupon collector's problem (from section 8.2) 8.3 Convolution with moment generating functions.	No class due to Thanksgiving.
13 11/28-12/2	8.4 Covariance and correlation.	Exam 2 in evening. Review class	8.4 Covariance and correlation. (skipped 8.5 The bivariate normal distribution)
14 12/5-12/9	9.1 Estimating tail probabilities. 9.2 Law of Large Numbers.	9.3 Central Limit Theorem. 10.1 Conditional distribution of discrete random variables.	HW 11 due. 10.1 Conditional distribution of discrete random variables. 10.2 Conditional distribution for jointly continuous random variables.
15 12/12-12/14	10.2 Conditional distribution for jointly continuous random variables	10.3 Further examples

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