**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

**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 (

**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 y

**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

Suggested Problems:

**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.

Suggested Problems:

**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.**

- Textbook for Math 161: Probability (Fall '06)
- Handout on Probability, Statistics, Proofs by Induction, Linear Algebra and Fourier Series. These notes are taken from my book (with Ramin Takloo-Bighash) An Invitation to Modern Number Theory.
- Definitions and common distributions.
- Sum of Poisson random variables.
- The Mean Value Theorem and Taylor Series.
- Approximations.
- The Cramér-Rao Inequality.
- The Median Theorem.
- Yes, Prime Minister: Manipulation Polls YouTube video
- Summary of Statistical Tests.
- Method of Least Squares
- Die Battles and Order Statistics.
- Differentiating Identities.
- Best of Seven: Coming back down 3-0
- Data for Best of Seven Series; (complete listing here)
- Pythagorean Won-Loss Formula for Baseball.
**Statistics Books involving Baseball by Michael Schell:**

**REFERENCES****:
**The following are some useful links:

- Textbook for Math 161: Probability (Fall '06)
- Library resources at Brown (and other links) (extremely extensive links)
- Mathworld (great site to look up information)
- Mathscinet (great site to look up published papers)
- JStor (great site to look up published papers)
- arXiv (archive of numerous mathematical work)