Math/Stats 425, Section 1, Fall 2004. Syllabus.


New! Follow up Courses in Probability and Statistics!
Instructor: Dan Burns
Office: 5834 East Hall
Phone: 7630152
Email: dburns@umich.edu
This is the home page for our section. This page is an update of the syllabus/homework assignment webpage from last year, together with a couple of links (see below) that may prove useful over the course of the term. For your convenience, here is the first day handout summarizing the course.
Written assignments for a given week will be due Wednesdays until the first hourly exam (October 1), after which they will be due Fridays. Check the assignments themselves below.
Midterm exams are in class exams, Friday, October 1 and Friday, November 5. These exams will be in our usual classroom. Since the class is followed by another class in the same room, these exams will be 50 minutes long, exactly.
The Term Paper can be substituted for the second midterm exam. Here is a list of suggested topics from past classes, which needs to be updated because there are more possible topics I can suggest now. You are strongly encouraged to take advantage of this option.
The Writing Supplement: 
If you are interested in taking the writing supplement to fulfill your
LSA junior/senior level writing requirement in the disciplines, send me an email right away. 
September 810, 2004  Read: Ross, sections 2.12.5 (read examples 5a5i first time through, others later)  Basic definitions and examples.  How do we model randomness? Frequentist approach; models and sample spaces. 
September 1317, 2004 
Read: Ross, sections 1.14. Write: Ch. 2, probs. #4, 7, 15, 28, 33. (Due Wednesday, Sept. 15) 
Counting (Combinatorics)  Useful for evaluating probabilities with finite numbers of outcomes. 
September 2024, 2004 
Read: Ross, sections 1.45, 3.13.2. Write: Ch. 1, probs. #15, 21, 22, 28, 29, 31; Ch. 2, probs. #39, 46, 52. (Due Wednesday, Sept. 22: Postponed until Friday, Sept 24, 2004.) 
More counting; multinomials ("manysided coins"); conditional probability. 
Multistage experiments, later outcomes conditioned on previous outcomes. 
Sept. 27  Oct. 1, 2004 
Read: Ross, sections 3.13.4. Write: Ch. 3, probs. 6,7,8,12,24,53,62. (Link is to written solutions for the highlighted problems.) (Due Wed. Sept. 29; NOT COLLECTED) 
Conditional probability; independence; Bayes's theorem. 
Some of the biggest concepts of the term! Exam Friday, October 1. 
Exam Prep for 1st MT 
Suggested Problems for Exam prep: Ch. 1 ST #2,4,6,10 Ch. 2 ST #2,4,7,11 Ch. 3 ST #3,8,12,19 ST = SelfTest problems (NOT collected; save for exam study) 
Moving into random variable, discrete RV, expectation. 
Random variable: another big concept for us. Just a glimpse before the first exam. 
October 4  8, 2004 
Read: Ross, sections 4.1  4.7 Write: Ch. 4, probs. #13, 21, 23, 26, 43. 
Variance; Bernoulli, binomial and Poisson RVs. 
Expectation (average);
Variance (and its square root, the standard deviation). These are everywhere important in statistics. 
October 11  15, 2004 
Read: Ross, sections 4.89; start 5.13. Write: Ch. 4, probs. #48, 50, 51, 53, 58. 
Examples of discrete RVs; beginnings of continuous RVs. 
Beginnings of continuous probability; now we have to use calculus as much as combinatorics and counting. 
October 18  22, 2004 NB: Monday is off! 
Read: Ross, sections 5.15, 5.7. Write: Ch. 5, probs. #3, 4, 9, 10, 26, 31. Note: These will be collected Wednesday, October 27. Also note: in class quiz, Friday, Oct. 22! (Link is to solution, 10/25/04.) 
Examples of continuous RVs; beginnings of normal RVs, DeMoivre's Theorem. Waiting times and exponential RV's. 
Big topic: normal RV. NB: Gil Stendig points out that problem #9 refers back to example 4b of Chapter 4, not 5b. Thanks, Gil! Refresh: two dimensional domains of integration; change of variables in integrals! 
October 25  29, 2004 
Read: Ross, sections 5.67, 6.13 Write: Ch. 5, problems 34, 35, 39, 40; Ch. 6, probs. 2, 6. 
Hazard Rate functions; change of RV's; Gamma distribution, lognormal distribution. 
Hazard rate functions a point of view in insurance and risk analysis; change of RV's enable random number generators to simulate general probability distributions; lognormal distribution used in finance to model markets. Sums are important for the Central Limit Theorem and Laws of Large Numbers. 
November 1  5, 2004 
Read: Ross, Chapter 6.4  6.5. No written problems collected this week; prepare for second midterm. 
Jointly distributed RV's; independence of RV's; sums of RV's; two dimensional events. N.B.: We did not treat Proposition 3.2 of section 6.3 in class, but it is quite important. Please have a look at it! 
Second MT Friday, November 5: Ross, Chapters 4, 5 and 6.13. Some selected "SelfTest" Problems for Preparation: Chap. 4: 3, 6 (long!), 8, 9, 13, 19; Chap. 5: 1, 6, 8, 13, 15, 18. Chap. 6: 1. 
November 8  12, 2004 
Read: Ross, Chapter 6.4  6.7; skip 6.8. Write: Ch. 6, #9, 15, 17, 32 (see Prop. 3.2), 42. Due: Monday, Nov. 15, 2004. Don't forget Ravi Vakil talk and Career Day Conference! 
Conditional distributions, discrete and continuous cases; functions of (several joint) RV's. 
Functions of RV's are important for simulating experiments with standard pseudorandom number generators. 
November 15  19, 2004 
Read: Ross, Ch. 6.6  6.7, 7.1  7.3, skip sec. 7.2.1 & 7.2.2. Write: Ch. 6, #39, 40, 45 Ch. 7, #8, 14, 18. 
Order statistics. Expectation, variance and covariance of joint RV's. 
``Order statistics" (6.6) important in statistics. The important new idea here is the covariance. 
November 22  24, 2004 
Read: Ross, Ch. 7.4  7.5, 7.7. skip 7.6 & 7.8. Write: Ch. 7, #21, 46, 47, 53, 55, 65. Due: Wednesday, Dec. 1. Click here for a course suggestion for W '05! Click here for many probability and statistics course suggestion for W '05! Happy Thanksgiving! 
Covariance. Conditional expectation. Prediction. 
These three topics very important in applications, especially statistics. 
November 30  December 3, 2004 
Read: Ross, Ch. 8.1  8.4. Write: Ch. 8, #3, 4, 6, 11, 14, 18, 20. Due: Wednesday, Dec. 8. 
``Multivariate"
normals. Moment generating functions. Background to Laws of Large Numbers. 
Multivariate normals are higher dimensional normals, with a
"shape". MGF technical but handy; used for the CLT. 
December 6  December 10, 2004 
Read: Ross, Ch. 8.1  8.6. Write: Start review problems for the exam; some suggestions may be found below. 
Central Limit Theorem (CLT), Strong Law of Large Numbers (SLLN). 
Term ends with a bang: two biggest results of the semester. LLN begins to justify the illdefined but popular ``law of averages". Central Limit Theorem is the "Universal Law of Errors". 
December 13, 2004 
Read: Ross, Ch. 8.5, 8.6. No written problems collected on this material. Finish up exam prep problems. Term projects due:

Estimating errors of approximations. Summary and review for final. 
Prepare your questions for review, as well as requests for posting solutions. 
GPS #1 Now includes solutions! 
Problems on basic probability modeling and conditional probability. 
Due: Wednesday, February 12, 2003. 
GPS #2 Now contains (some) solutions! 
Expectation and continuous RV's  Due: Wednesday, March 19, 2003. 
GPS #3
Now contains solutions! 
Independence (again); the ``Poisson Process"; Monty Hall! 
Due: Friday, April 11, 2003. 
Ross, Chapter 4  Problems # 24, 27, 45, 49, 54, 62, 67. 
Be sure to understand basic ``modeling", i.e., what ``kind of randomness" a particular shape of probability mass function models. 
Ross, Chapter 5  Problems # 6, 8, 11, 14, 29, 34 39. 
Same comment as above; also be sure how to describe ``events" involving RV's in terms of regions of integration. 
Ross, Chapter 6  Problems #12, 13, 25, 44, 51. 
Joint distributions: know well ``events" as regions of integration; sums of RV's and their expectations. 
Ross, Chapter 7  Problems #24, 34, 41, 42, 47, 56. 
Properties of expectation and conditional expectation and
variance. Some of these problems in chapter 7 can be quite long. Don't expect to do all of those suggested in a short period. Look them over and try to analyze how to procede. Then work out all the details on a couple of examples. Be sure to look at the ducks and the elevators (#55 and #56). 
Ross, Chapter 8  Problems #18, 20. 
LLN and CLT (jargon you should know!)
20 is a bit theoretical for the level of 425; 18 is not. 
Announcement: Final Exam 
Tueesday, December 21, 2004, 10:30 AM  12:30 PM. Room TBA. 
Note room TBA!  ``Can't wait!" 