... An Honors and Graduate Course at Wharton

Course Blog Fall 2013

Last Minute Questions. Etc.

Feel free to e-mail me any time. I won't be back to the office before exam day, but I can help with many questions via email.

About the Exam

The exam has been written. It has eight questions, one of which is a total warm-up question which everyone will crush. The other questions are straightforward if you are well prepared. The only possible exception is the last question which has some challenge to it; it's not hard but the answer is charming. Most of the questions do have multiple parts. You may be able to finish quickly, but the smart thing is to use all of the time that you have. Hang in there and check your work until you have zero doubts. Most problems are of the form "Calculate ...." but a couple are of the form "Explain Why or Why not.."

Do also see the instructions for the exam. BTW, your "formula sheet" can have formulas on both sides of the sheet.

Time and Place: Friday Dec 20, 9AM-11AM, 345JMHH

Please note this course is Stat 510 and Stat 430H. Our exam is at a different time and place from the exam for Stat 430. Please cast your own eyes on the Exam Time and Room List.

Practice Exams

If you Google "Probability Final Exam" you find a great number of old final exams and proposed practice exams. The only issue is how to calibrate these with the expectations of this honors course. Here are some examples with my calibration:

  1. UML 431 This is a good exam. It is has more on the normal approximation than I would have included, but it has some good conceptual problems --- ones that measure understanding without too much computation.
  2. LSA 425 Another good exam. A few of the problems are too easy for us, given that we have spent a fair amount of time discussing the gamma distribution.
  3. USC 407 This is pretty well calibrated, though perhaps a little on the easy side.
  4. UIUC 408 This is again a reasonable exam. It's perhaps a little too routine.
  5. Stanford 116 Ah, a course I taught many times. This is a decent exam, though not quite the honors version. Problem 3 is too routine, but Problem 6 is nice.

If you work through these exams and if you review your notes to see where we have gone beyond the usual introduction (e.g. Strong vs Weak LLN, Cauchy distribution, lots on Gamma, Jensen's inequality, Record random variables, Poisson processes, Le Cam's inequality, Box-Muller method, etc) then you will do fine. I will also give examples in class of some more "creative" problems --- the starred problems of our text are of the creative kind, though typically a bit too long for an exam.

Be sure to bring your questions to class; if something is an issue for you, it is likely to be an issue for other people too.

Office Hour Changes and Special Times

I will not have office ours on Wednesday November 20, but I will have special office hours on Friday November 22 from 1:15pm to 2:45pm. I will have to leave at 2:45 sharp to attend a Ph.D. oral exam.

Homework? No More ... but ..

Formally, there won't be further homework, but there is plenty of work to be done. We only have five more classes, but these are broken up over four weeks. This gives one a mountain of time to integrate the material of Chapters 1 through 6, deepen our coverage of the Poisson process and Poisson approximation, revisit some classical problems (Birthday, Coupon Collector, Craps), and learn a bit about martingales --- a rich model of a fair game that is also perhaps one of the most powerful tools in probability.

Great TED Talk on Memory

Last week I mentioned a mnemonic for recalling the English royal families, and it offered a brief moment of quizzical amusement, but there is a larger point. Memory training is an honestly fascinating topic that almost any student may find useful.

At a minimum it is worth scooping out. There is a great TED Talk by Joshua Foer that gives a hilarious introduction. I can't imagine how to get more out of an investment of 10 minutes of your time. The key concept is "elaborative encoding", and it works.

Homework Due Monday November 18

Read through the problems pp 292--294 in the text. These problems are all "starred" but it is important to understand them. You should particularly note that problem 16 is a version of the First Borel Cantelli Lemma. The Wikipedia has a useful discussion of the BC lemmas; give it a read and compare our discussion of BC-2 with the one given there.

Read Chapter 6 Sections 1 and 2. Do problems 1, 2, 3, 10, and 11 at the end of Chapter 6.

For review and reinforcement, you should also read Grinstead and Snell Chapter 3. This deals with material that we covered long ago and it offers a slightly different perspective.

Good News: We will not have further written homework after the assignment that is due November 18.

Still, I will make further suggestions for review. Moreover, after completing our work with Chapter 6, we will cover some material on martingales (lectures only) and some material on random walks (see Chapter 12 of Grinstead and Snell.

Craps: All the Rules

Craps is the most popular casino game in the US after blackjack. It is not "horrible" though you will lose with brutal certainty, just not as fast as in some other games. One of the rites of passage in a basic probability course is to calculate the expected value of a one-dollar bet "on the pass line" at craps. The venerable Wikipedia will give you the rules; go to the section on bets on the pass line. Some financial firms have used this as an interview question.

Homework Due Monday November 11

Complete your reading of Chapter 5. Do problems 8, 9, 10, and 11 at the end of the chapter.

A Public Service Announcement for a Former Student

"Please join representatives from Citi’s Sales and Trading team on Thursday, November 7th, to participate in a Sales and Trading Workshop that will cover Sales and Trading basics, current topics in the market, common pitfalls on the road to a summer internship, and tips for how to succeed in the Sales and Trading Analyst program. Citi traders and salespeople will be available following the event for networking. Details as well as upcoming key dates are listed below. 
To RSVP, please email your resume to julie.blair.samuel@citi.com prior to the session.  We look forward to meeting you!
S&T 101 Workshop:
Date: 11/7/2013
Time: 8:00 - 9:00 pm
Location: Jon M. Huntsman Hall in the Wharton School Building, Room G-55
*This session will be most useful to Sophomores and Juniors interested in Citi’s Summer Analyst program but Master's and Ph.d. students will also be considered.

Homework (Due Monday November 4)

Read Chapter 5, Sections 5.1, 5.2, and 5.3. Review Chapter 4 for mastery.

Read the MIT Lecture on the Gamma distribution. This has a bit more in it than we have covered, but it is well written and the notation is consistent with the notation we have used in class. For example, the lecture also covers the F-distribution which we won't engage for a few weeks, but it also deals with the Chi-squared which we will discuss this week.

Some articles on the web on the Gamma are defective; a nice reminder to read with a skeptical eye. One article on "AP Statistics" was brutally flawed; I won't grace it with a link.

Do problems 41, 42 and 43 (Chapter 4, page 260) and do problems 1, 4, and 5 at the end of Chapter 5. (Total of 6 problems)

Homework (Due Monday October 28)

Complete reading of Chapter 4, including the worked (or starred) problems.

Do the problems 24, 29, 32, 33, 34, 35, 36, and 37 at the end of Chapter 4.

Homework (Due Monday October 21)

Continue reading Chapter 4 through Section 4.4. Also fill in any cracks in your earlier reading, e.g. read Problem 35 page 198--199. The "stared problems" contain more specialized information than the main text, but much of this specialized information is important, even though we can't cover it all in class.

Problems: Chapter 4 Problems 5, 12, 17, 18, 19, 22, 23.

Finally: Prepare yourself as if you were going to have a mid-term. Specifically, review all of the material in through Section 4.4 then (1) create a mind-map of the concepts and facts and (2) make a list for yourself of "canonical problems".

This review material is not to hand in but for you to help organize and integrate the material. I'll give examples in class.

Human Talent and "Let X..."

Work on the Salisbury Cathedral was begun in 1220, and you can bet your boots that there was a lot of planning that went on before any foundations were dug. Look on this mighty edifice and ponder: No one in England at the time would have any understanding of a sentence that began, "Let X denote the unknown quantity ..." In fact, if you had that thought, you'd be wise to keep it to yourself.

Supplemental Text

If you would like to read another text that is roughly at the same level as our text, I recommend that you look at the text by Grinstead and Snell. It gives a second view, and it really is quite good. It is also freely offered on-line by the authors; guilt free downloading.

Homework (Due Monday October 14)

Complete your reading of Chapter 3. By now you should start to feel comfortable with the notations for joint and conditional densities. Read Chapter 4 sections 1 and 2.

Problems: Chapter 3 problems 19 and 34; Chapter 4 problems 1,2, and 3.

For "cultural background" take a peek at the Wikipedia article on the Beta Function. As with any encyclopedia article it has more than you need, but it has stuff that is useful to have seen.

In Memoriam: Tom Clancy

Novelist Tom Clancy died on October 1 at the young age of 66.

In 1989 I wrote a paper that was motivated by Clancy's first novel, The Hunt for Red October. The paper is not hard; it would be easily understood by graduates of our class. Nevertheless, it seems to me that it nicely illustrates how one can profitably "follow a thread."

Check out my first few lines.

Incidentally, if you are at Princeton and you publish a paper called "Models for Managing Secrets" you can expect to pick up some attention from the world's three letter agencies. It's part of the Wilsonian tradition --- and the attendant back splash.

Homework (Due Monday October 7)

Read Chapter 3 though at least Section 3.3. Please also read though the solved problems for these sections. These problems contain interesting and important material.

Do the following problems from Chapter 3: 1,2,5,6, 8, 11, 12, and 13.

Homework (Due Monday September 30)

Complete your reading of Chapter 2. Give particular attention of Sections 2.5-2.6 but give some time to review of previous reading. Also, in your reading, don't forget to read the problems. Problem 28 is very nice and the solution is given in the text. You certainly don't have to solve every problem that you read, but if you find a problem that interests you, go for it. That's the most enjoyable kind of learning.

Do the following problems: 26, 31, 32, 38, 39, 40, 41.

Homework (Due Monday September 23)

Complete a careful reading of Sections 2.1-2.4 of the text. Do problems: 1, 2, 3, 6, 7, 16, 17, 21, and 25 of Chapter 2. Note that Problem 21 was discussed in class. When you answer "how much would you pay" this is a question about YOU. There is no "right answer" so search your soul and see what you really would pay. Also note that Problems 6 and 7 are not routine; you may have to do some numerical exploration to get a good answer.

Day Three

We're now hitting the natural stride of the course with an honest two days work per week. Today we'll discuss independence --- a central concept of probability. We'll then take up the technology of combinatorial counting. Long ago combinatorics and probability theory were almost synonymous. Now the two field have many distinct elements, but still there are plenty of elements in common.

Do problems 52, 53, 54, 55, 56, 57, 58,59, and 60. This is a serious chunk of work, but the only way to learn how to solve combinatorial problems is to solve them. You should also begin reading of Chapter 2. During the week, we'll cover perhaps the first third of this chapter.

Corrected Homework: Complete your reading of Chapter 1. We won't engage the section on counting just yet, but it pays to have an advanced look. Do problems 16, 19, and 22 of Chapter 1. Note: In problem 19 the task is really to make the identifications that show that the formulas of the problem are consequences of general formulas of the text. These problems are due Monday September 9.

Day Two

For a warm-up, I'll discuss the famous formula for the "ruin probability" when one plays an unfair game. This discussion should save you several thousand dollars some weekend of your future life.

It will be much later that we will see why this formula is true, but for the moment we can still extract some life lessons from it. There is a message about Casinos, but we'll also engage some philosophical question: What are the ways one "understands" a formula? What are the purposes of "knowing" a formula? These are more nuanced question than one might guess.

We'll go on to discuss conditional probability. This will lead in turn to Bayes's rule and a massively instructive calculation that may change your view of diagnostic tests.

Finally, we take up the idea of independence. After the basic definitions, this is the single most important concept in probability theory.

Venn Diagrams

Venn diagrams are not the deepest things in the world, but sometimes they do help one see something a bit more clearly. We'll use them to get started on the inclusion-exclusion principle, though they can't be used for much more than the first step. Some Venn diagrams acknowledge their pretentiousness. You may like the one below if you have seen the classic movie Gone with the Wind, where Rhett says to Scarlett, "Frankly, my Dear, I don't give a damn." Scarlett being Scarlett, pulls through: "Tomorrow is another day."

Frankly, My Dear,

Important Textbook Notice

Be sure you get the correct text. You need Bertsekas and Tsitsiklis (2nd Edition). There are copies in the bookstore (25 more arrived today). Some copies may be stacked under other courses but look under Stat 510-430H.

There is another text that is available on-line and which you can use to supplement your reading of our text. This is the book by Grinsteadt and Snell. There is often a benefit to reading the "same thing" said two different ways.

Day One

After a brief top-down view of the course objectives, we'll get straight to work. The basic plan will be to cover the Sections 1.1, 1.2, and 1.3 of the text. Specifically, we'll cover the basic ideas of a probability model, solve some basic problems, and find the probability that Romeo and Juliet meet on a particular evening. We may even look at the difference between flipping coins and spinning them. This speaks to the general idea of being attentive to distinctions.

Your first homework assignment is to do problems 5, 6, 14, 17, and 18 of Chapter 1. This homework will be due on Wednesday September 4 at the beginning of class. There will be homework assuagements every week, and timely attentiveness to these assuagements is the surest way to do well in the class.

Please also review the course policies. I count on our little community to provide collective compliance with the policies, so if you see someone fall into an iPhone coma please take appropriate action so I won't have to disturb the class (and seem like a heal).

You will also want to look at the course syllabus for information about grading, homework, the midterm, and the final exam.