MATH  162:  MATHEMATICAL  STATISTICS: MWF 2:00 - 2:50pm in BH157

FINAL EXAM (according to Banner, though you have to REALLY navigate well to see this): Friday, May 9th, 9am in BH 157.

CLICK HERE FOR LIST OF PROJECTS

Professor Steven Miller (sjmiller AT math.brown.edu), Kassar House, Room 210, 401-863-1123
Office Hours: before class MWF, Kassar 210

COURSE DESCRIPTION: Central limit theorem, point estimation, interval estimation, multivariate normal distributions, tests of hypotheses, and linear models. Prerequisite: MA 161 or permission of the instructor. More specifically, the goal is to develop the tools and techniques necessary to see if numerical evidence supports conjectured relationships. Examples will be drawn from various parts of mathematics (number theory, random matrix theory, continued fractions, probability theory) as well as real-world problems. There will be a strong emphasis on the theory behind the statistical tests. Please read the relevant sections before class. This means you should be familiar with the definitions and what we are going to cover. There are numerous Handouts to supplement the text (John E. Freund’s Mathematical Statistics with Applications (seventh edition), Pearson / Prentice-Hall, ISBN 0-13-142706-7; note some of the text is available online by clicking on the link). Calculus is a pre-requisite for this course; you will not be able to do many of the homework and exam problems if you cannot do calculus. There will be a calculus diagnostic assignment in the first week of class. If you fail you must complete an extensive calculus assignment to remain in the course.

EXAMS / GRADING / HOMEWORK: I encourage you to work in groups, but everyone must submit their own HW assignment. HW is to be handed in on time, stapled and neat -- late, sloppy or unstapled HW will not be graded; please try and do all problems in order. Please show your work on the HW and exams (otherwise you risk getting no credit). There will be one or two midterms, a short paper (around 5-10 pages, using statistics to analyze an appropriate problem of your choice; more details will be provided during the semester), and a final. Homework 20%, Paper 10%, Midterm(s) 30%, Final 40%. Exams are black tie optional.

HOMEWORK

Please spend at least 1 hour a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

_*^*_* BELOW IS A TENTATIVE SYLLABUS, AND IS SUBJECT TO CHANGE. _*^*_*

Week One (1/23 - 1/25):
Read: Skim Chapters 1 - 4 of the textbook and Chapter 1 of the additional course notes below (this should mostly be review material from Math 161). Chapter 7: 7.1 - 7.4 (should be review).
Suggested HW (you do not need to turn this in):  probability review problems        calculus review problems        many calculus review problems     Regardless of what the .pdf files above say, you do not need to hand them in; you are on the honor system to make sure you can do all these problems. If you have any difficulties with them, please email me.

Week Two (1/28 - 2/1):
Read: Chapter 7: 7.5 - 7.6, special probability distribution handout, sums of poisson random variables; Chapter 8: 8.1, 8.2, 8.4.
HW: Due Monday 2/4:  Page 137: #4.8; Page 146: #4.23; Page 237: #7.1, #7.4; Page 252: #7.11, #7.17; Page 258: #7.45, #7.46; Page 272: #8.3, #8.4.
Extra Credit: Due Monday 2/11: Find a probability distribution such that the inequality in Chebyshev's Theorem is an equality for all positive integers, or show such a probability distribution does not exist.
Suggested Problems: Page 272, #8.13.

Week Three (2/4 - 2/8):
Read: Chapter 8: 8.4 - 8.8 (just skim 8.6, won't be on any test) and handout on the the Median Theorem and Wallis' Formula. If time permits we will start 10.1 and 10.2.
HW: Due Wednesday 2/13:  Page 283: #8.21; Page 291: #8.58a, #8.61a, #8.70, #8.73; Page 285: #8.42; Page 288: #8.44, #8.45; #8.57. Hand in project statement.
Extra Credit: Due Wednesday 2/13: Find a probability distribution such that the inequality in Chebyshev's Theorem is an equality for all positive integers, or show such a probability distribution does not exist.
Suggested Problems: Page 288: #8.56 (HINT: there are often two ways to do the change of variables -- depending which way you choose can make the algebra harder. Use Theorem 7.2 on page 248. Let y₁ = w₁(y₁,y_n) and p = w₂(y₁,y_n) = F(y_n) - F(y₁). The Jacobean J = f(y_n). This implies g(y₁,y_n) = h(y₁,p)f(y_n), and then substitute). ALSO SHOW g(p) is a probability density.

ALMOST SURELY THE HW AND READING FROM THIS POINT ON WILL CHANGE AS CLASS PROGRESSES.

Week Four (2/11 - 2/15):   Read Method of Least Squares. No class on Monday; use the time to think about your project and hand in the summary on Wednesday. CLICK HERE FOR LIST OF PROJECTS
Read: Chapter 8: 8.7 - 8.8 and handout on the the Median Theorem; Chapter 10: 10.1 - 10.3 and the handout on The Cramér-Rao Inequality.
HW: Due Wednesday 2/20:   Page 293: #8.76, #8.80. #8.86, #8.89;   Page 325: #10.1, #10.11, #10.12. On a separate piece of paper, from the handout on the the Median Theorem do Exercise 3.2 and read Exercise E.5; note: if you handed in your project statement on time, do not hand in Exercise 3.2.

Week Five (2/20 - 2/22): NOTE: There is no class on Monday. CLICK HERE FOR LIST OF PROJECTS
Read: Chapter 10: 10.3 and the handout on The Cramér-Rao Inequality; Chapter 14: 14.3, Method of Least Squares; Chapter 10: 10.4, 10.6, 10.7, 10.8.
HW: Due Wednesday 2/27:  Page 325: #10.16, #10.19, #10.20; Page 350: #10.80.

Week Six (2/25 - 2/29): CLICK HERE FOR LIST OF PROJECTS
Read: Chapter 10: 10.8; Chapter 11: all sections
HW: Due Wednesday 3/5:  Please hand in an approximately one page writeup of your project. It should briefly say what you are going to test and what data you'll collect (and how you plan on doing it). Don't worry if you don't know a statistical test to use. The point of this is to make sure you know what kind of data you need to gather so you can start doing it. Many of you wrote detailed descriptions for the first project assignment; if you did that you can just resubmit that (others wrote just a sentence or two, and I want to make sure everyone has a good sense of what they need to do).
HW: Due Friday 3/7:  Page 325: #10.21, #10.22, #10.23; Page 335: #10.35; use BOTH the method of moments AND the method of maximum likelihood to find the parameters of Uniform[a,b], given sample data of x₁ = 1, x₂ = 3, x₃ = 2.5, x₄ = 1.5. Page 370: #11.20,  #11.26; Page 361: #11.1; Page 371: #11.35.
Suggested Problems: Page 370: #11.21, #11.27; Page 362: #11.4, #11.5; Page 373: #11.59.

Week Seven (3/3 - 3/7):  NOTE: Review session Monday, 3/3 from 8-8:50am in Kassar 105. NOTE: midterm will be Wednesday, 3/5, in Kassar 105. Choose any 90min block from 7am to 9am. The midterm will cover all material up to and including chapter 10, the handouts and the supplemental material covered in lectures (but not chapter 11).
Read: Chapter 12:  12.1, 12.2, 12.4.
HW: Due Wednesday 3/12:  work on projects
Suggested Problems:
 

Week Eight (3/10 - 3/14):  On Friday Susan Silverman will talk on "Insurance Statistics: It's all in the data". She has a Sc.B. in Mathematics from Brown and took MA 162. She's a fellow of the Society of Actuaries and a Member of the American Academy of Actuaries. She worked for John Hancock for many years in the areas of Underwriting, Group Pension, Group Insurance and Individual Life Insurance, and currently teaches actuarial science at Boston University. We will start class 10 to 15 minutes late so people can go to the beginning of π day.
Read: Chapter 13: 13.1, 13.2, 13.3, 13.4, 13.5, 13.6, 13.7, 13.8, 13.9.
HW: Due Wednesday 3/19:  Page 383: #12.1ab, #12.6, #12.11; Page 395: #12.32; Page 409: #13.2; Page 424: #13.19, #13.20, #13.27, #13.30.
Suggested Problems: Page 343: #10.56,  #10.63, #10.66; Page 370: #11.25, #11.33, #11.35; Page 424: #13.25.

Week Nine (3/17 - 3/21):  NOTE: optional second midterm will be 3/19 in class. There will be no class on Friday: use the time to start gathering your data and work on your project. You should have the project done by mid to late April.
Read: Chapter 13: 13.9. Chapter 14: 14.1, 14.2, 14.3.
HW: Due Monday 3/31: Page 417: #13.13; Page 430: #13.77, #13.78, #13.83.

Week Ten (3/24 - 3/29):  BREAK: NO CLASS: Continue reading and working on HW / project, read handout on differentiating identities.

Week Eleven (3/31 - 4/4):  NOTE: no class on Friday. Use the time to continue to work on your projects (which you should have mostly done within a week or two). There will be a review class on Wednesday from 8 - 8:50am in Kassar 105 (we'll discuss the recent exam, questions about material, and if time permits any questions on your projects).
Read: Handouts: Chapter 14: 14.1, 14.2, 14.3, 14.4, 14.5. Review Sections 3.6, 3.7, 4.7, 4.8 and the class notes on Least Squares
HW: Due Wednesday 4/9:  Page 113: #3.74; Page 158: #4.47ab; Page 451: #14.17, Page 472: #14.42, #14.50 (this is known as a power law).

Week Twelve (4/7 - 4/11): 
Read: Chapter 14: 4.7, 4.8; Chapter 16: 16.1, 16.2, and differentiating identities (pdf) handout
HW: Due Wednesday 4/16: Page 529: #16.2 (write the proof carefully); Page 544: #16.16, #16.17 and the following problems:

• [Newton's Method Problem] We say x_0 is a fixed point of a function h if h(x_0) = x_0. Let f be a continuously differentiable function. If we set g(x) = x  - [f(x)/f'(x)], show a fixed point of g corresponds to a solution to f(x) = 0; note g(x) is the function that arises when we do Newton's Method. Assume that f:[a,b] to [a,b] and there is a C < 1 such that |f'(x)| < C for x in [a,b]. Prove f has a fixed point in [a,b]. Is the result still true if we just assume |f'(x)| < 1? Fixed points have numerous applications, among them showing optimal strategies exist in n-player games.

• [Differentiating Identities Problem] Consider the exponential distribution with parameter b: Prob(x) = (1/b) exp(-x/b). By differentiating identities (DO NOT USE ANOTHER METHOD), find the mean and the variance. FIRST HINT: We have Int_{x: 0 to +oo} (1/b) exp(-x/b) dx = 1; it is technically convenient to rewrite this as Int_{x: 0 to +oo} exp(-x/b)dx = b. SECOND HINT: modify the result of the first differentiation so the second differentiation is easy.

• [Optional Reading] If you are interested in winning at blackjack: Thorp's Article and Thorp's Book: Bringing Down Vegas.Thorp's Book: Beat the Dealer. NOTE: I do not advise going to Vegas and using these strategies -- these are simply interesting probability exercises.

Week Thirteen (4/14 - 4/18):  Optional exam in class on Monday, 4/14 (details about the exam announced in class).
Read: Chapter 16, 16.3, 16.6, and differentiating identities (pdf) handout. Chapter 15: 15.1 - 15.2.
HW: Due Monday 4/21:Page 540: #16.13, #16.14; Page 546: #16.29, #16.34 (get someone outside of the class who doesn't know any of our randomness tests to generate the 100 tosses). Exercises 3.3 and 3.4 of the and differentiating identities (pdf) handout (page 8).

Week Fourteen (4/21 - 4/25): 
Read: Chapter 15: 15.1 - 15.2
HW: Due Wednesday 4/30. Page 444: #14.1, #14.3, #14.10, #14.15, #14.16; Page 451: #14.17, #14.18; Page 457: #14.30; Page 493: #15.2. Applied Problems: Page 478: #14.67; Page 513: #15.17, #15.19.

FINAL EXAM (according to Banner, though you have to REALLY navigate well to see this): Friday, May 9th, 9am. There will be an optional exam as well (probably either immediately after the final or on the last day of class).

Please spend at least 1 hour a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

ADDITIONAL COURSE NOTES: Note the textbook (or at least part of it) is available online HERE.