Math 431 - Introduction to Probability Theory

Fall 2012 - Lecture 2

Meetings: TR 11:00 AM - 12:15 PM, Van Vleck B115
Instructor: Benedek Valkó
Office: 409 Van Vleck
Phone: 263-2782
Email: valko at math dot wisc dot edu
Office hours: Monday 3:30-4:30PM, Tuesday 1-2PM or by appointment

I will use the class email list to send out corrections, announcements, please check your wisc.edu email from time to time.

Course description

Math 431 is an introduction to the basic concepts of probability theory, the mathematical discipline for analyzing and modeling uncertain outcomes.
Probability theory is applied in several disciplines (e.g.~natural sciences, social sciences, economics, engineering) so the course can be valuable in conjunction with many different majors.

Textbook: Jim Pitman: Probability, Springer


Prerequisites

The basic skills needed for Math 431 are calculus (including multivariable calculus) and basic set theory. Success in this class will also require the ability to think/reason abstractly.

Course content

The course consists of Chapters 1-6 from Pitman's book. The main topics covered in these chapters are

Evaluation

Course grades will be based on home work assignments and quizzes(20%), two midterm exams (20% each) and the final exam (40%).
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.

Exam dates

Note that the midterm exams are evening exams! There will be no make-up exams or quizzes. The use of calculators is not permitted on tests, quizzes and during the final exam. You may use calculators for your homework assignments.
You can find 431 exams from previous years at the math library exam reserves. Note that these exams came from a different textbook.

Quizzes

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

Homework assignments

We will have weekly assignments posted on the course website (here).
Homework must be handed in by the due date at the beginning of the class. Late submissions will not be accepted. Group work is encouraged, but you have to write up your own solution.
You can use basic facts from calculus, linear algebra and probability and also the results we cover in class. If you use other literature for help, cite your sources properly. (Although you should always try to solve the problems on your own before seeking out other resources.)
The assignments will contain bonus problems. These will be more challenging than the regular homework and will have more flexible deadlines. The bonus problems should be submitted separately from the usual homework assignment.

Instructions for the homework assignments:

Homework


 Problems
 Due Date
HW1
 1.1: Problem 6 (page 10)
1.2: Problem 2 (p. 18)
1.3: Problems 4, 6, 8, 11, 14 (p. 30-32)
1.4: Problems  4, 6, 10 (p. 45-46)
 September 13, Thursday (beginning of class)
HW2
 1.5 :  2, 4 (p. 53)
1.6 : 1, 4, 8 (p. 70-71)
Review exercises: 4, 6 (only the independence of the complements) 8 a), b) , 10 a), 11 (p. 74-75)
neither
 September 20, Thursday (beginning of class)
HW3
 2.1: 2, 4, 12 (p. 91-92)
2.2: 1, 4, 6 b), 9 a), c)  12, 13 (p. 108-109)
You may use the table for Phi(x) in the book (p. 531)
 September 27, Thursday (beginning of class)
HW4
 2.4: 2, 4, 6, 9 (p. 121-122)
2.5: 2, 4, 8, 9 (p. 127-128)
 October 4, Thursday (beginning of class)
HW5
 3.1.  4, 6, 12, 22 (p. 158-160)
3.2  8, 10, 14, 21 (p. 182-184)
 October 16, Tuesday (beginning of class)
HW6
 3.3 4, 8, 13, 20 (p. 202-205)
3.4 4, 8, 10, 12, 24 a), b) c) (p. 218-221)
Hints:
3.3.20 b) Use the Central Limit Theorem
3.4.8 Use the `Craps Principle' on p. 210
3.4. 24 You can use the fact that 1+1/4+1/9+1/16+... is finite. (It is equal to pi^2/6.)
 October 23, Tuesday (beginning of class)
HW7
 3.5 3, 9, 15, 18  (p. 234-236)
3.6 2 c) d), 3 a) e), 6 a) b) (p. 243-244)
4.1 2, 7 (p. 276)
4.2 10 a) (p. 294)
Hints:
3.5.3 It seems that the numerical answer in the book for part a) is a little bit off.
3.5.18 is perhaps the most interesting problem of the set. You need only basic properties of Poisson we discussed in class. For (a) look at the first few Xn to see the pattern, make a conjecture for the general Xn, and verify your conjecture by induction.
4.1.7 Find the standard deviation first using the standard normal distribution table in the back.
November 1, Thursday (beginning of class)
HW8
 4.2 5, 12 a), b) (p. 293-294)
4.4 2, 5, 6, 10, a), d)  (p. 309-310)
4.5 2 a), 6 (p. 323.)
 November 8, Thursday (beginning of class)
HW9
 5.1 1, 7, 8 (p. 344-345)
5.2 1, 3, 8 a) (p. 354)
November 20, Tuesday
(beginning of class)
HW10
 5.3: 3, 5, 8 (p. 367-368) 
5.4: 1, 13  (p. 384-385), and this problem:
find the density of X+Y when X and Y are independent and exponentially distributed with distinct parameters λ ≠ μ.
6.1: 5(a)  (p. 399), and this problem:
Using the notation of Example 4 from p. 213 (negative binomial),
a) find the joint probability mass function of (T1,T2)
b) find the conditional probability mass function of T1, given that T2=n.
 December 4., Tuesday
(beginning of the class)
HW11
 6.2: 1, 6(a)(b), 12 (p. 406-408) 
6.3: 2, 4, 11 (p. 426-427)
6.4: 1, 5, 21 (p. 445-447)
 December 13, Thursday
(beginning of class)


Practice problems for the second midterm

You should be able to solve all of the following problems (some of these appeared on the homework assignments):

3.1: 1-12, 15-17, 22 (some more challenging problems: 13, 14, 18, 19, 24)
3.2: 1-12, 15-18 (harder: 19-22)
3.3: 1-5, 7-9, 12-20, 23 (harder: 24-30)
3.4: 1-12, 15, 17, (harder: 13, 14, 18, 19, 20, 21, 24)
3.5: 1-17 (harder: 18, 19 a-c)
3.6: 1-9 (harder: 10-16)
Ch 3 Review: 1-10, 13-17, 19, 20, 22, 25-27, 34 (harder: 12, 18, 21, 23, 24, 29, 30-32)
4.1: 1-11 (harder: 12-15)
4.2: 1-11, 12 a, b
4.4: 1-10
4.5: 1-8 (harder: 9)
Ch 4: Review: 1-16, 18-19, 21-26 (harder: 27-30)

The second midterm covers everything that we covered in class up to (and including) Section 4.5. (Note: if we skipped a section in class then you don't need to know that part of the material.)

Practice problems for the final exam

You should be able to solve all of the following problems (some of these appeared on the homework assignments):

5.1: 1-8 (harder: 9)
5.2: 1-5, 6 a, b, 7, 8 a, b, 9, 11, 12, 13, (harder: 10, 14-20)
5.3: 1-12, (harder: 13-16, 18)
5.4: 1-4, 6, 13,  (harder: 5, 7-12, 14-15, 19)
Ch 5 Review: 1-8, 10-17, 20-21, 22 a),   (harder: 9, 18, 19, 23-25)
(Problems from Chapter 6 will be posted soon.)
6.1: 1-8 (9-10)
6.2: 1-13, (15-16)
6.3: 1-12 (13,15)
6.4: 1-18, 21-22
Ch 6 Review: 1-11, 13 a), 18 a)

How to succeed in this course

It helps immensely if you keep up with the material during the semester by reading the textbook and following the lectures. I will post section numbers of the textbook for upcoming lectures, it is beneficial to read ahead.
Although reading the textbook is important, the best way to master the material is through problem solving. This of course includes the homework assignments, but you shouldn't stop there, there are more than enough practice problems in the text book. (If you run out of problems to solve, I'm happy to supply more!)
For some students studying in groups helps a lot, let me know if you need help with setting up a study group.
If you feel lost, do not hesitate to ask for help. I'm available in my office hours (see above) or by appointment. Do not wait until the end of the semester if you think you are in trouble!

Schedule


 Covered topics
 Suggested reading for next week
Week 1.
(9/4, 9/6)
 Equally likely events, axioms of probability, conditional probability and independence
Sections 1.1-1.4
 Sections 1.5-1.6. 2.1-2.3
Week 2.
(9/11, 9/13)
 Bayes' Rule, sequences of events, birthday problem, independence of multiple events
Sections 1.5-1.6, 2.1
 Sections 2.1-2.3
Week 3.
(9/18, 9/21)
 Geometric distribution, binomial distribution, normal approximation
Sections 2.1-2.3
 Sections 2.3-2.5
Week 4.
(9/25, 9/27)
 Normal approximation (cont.), Poisson approximation, random sampling
Sections 2.3-2.5
 Sections 3.1-3.2
Week 5.
(10/2, 10/4)
 Random variables, expectation,
Sections 3.1-3.2
 Sections 3.2-3.4
Week 6.
(10/9, 10/11)
 Properties of expectation, indicator random variables, variance, Markov and Chebychev inequalities
Sections 3.2-3.3
MIDTERM EXAM
Sections 3.4-3.6
Week 7.
(10/16, 10/18)
 Discrete distributions, Coupon collector, Poisson distribution, Poisson scatter theorem
Sections 3.4-3.5
 Sections 4.1-4.2
Week 8.
(10/23, 10/25)
 Exchangeable distributions,  Continuous distributions, normal, uniform and exponential distributions, the Poisson arrival process
Sections 3.6, 4.1
Sections 4.4-4.5
Week 9.
(10/30, 11/1)
 Gamma distribution, Change of Variables, Cumulative distribution functions
4.2, 4.4, 4.5
 Sections 5.1-5.2
Week 10.
(11/6, 11/8)
 Continuous joint distributions, uniform distribution in higher dimensions, joint density function
5.1-5.2
 Sections 5.3-5.4
Week 11.
(11/13, 11/15)
 Review session, operations on random variables, convolution
5.2, 5.4
MIDTERM EXAM
Sections 5.3-5.4
Week 12.
(11/20)
 Convolution of normals, linear combinations of independent normals, Rayleigh distribution
5.3
 Sections 5.3, 6.1-6.2
Week 13.
(11/27, 11/29)
 Conditional distribution for discrete random variables, conditional expectation with respect to an event and a random variable
6.1-6.2
 Sections 6.3-6.4
Week 14.
(12/4, 12/6)
 Conditional distribution of continuous random variables, conditional expectation, covariance, correlation
6.3-6.4
 Section 6.5
Week 15.
(12/11, 12/13)
 Covariance, Bivariate normal
Review