Math 430, Introduction to Probability

Fall 2009

Information

Contact information
Office DRL 3C7
Office hours By appointment, or just stop by
Phone 86219
Email guffin.math.upenn.edu
Course information
Location DRL 3C4
Times MWF 1pm-2pm
Problem Sessions Monday 3pm-4pm, DRL 4C4
Tuesday 5pm-6pm, DRL 3C4

Course handout

Course Description

Probability is the study of random events. Math 430 is a one-semester introduction to concepts and methods in probability for sophomores, juniors, and seniors in mathematics, the physical and social sciences, engineering, and computer science. We will cover a broad range of ideas in probability as well as techniques necessary for a firm understanding of the subject.

int getRandomNumber(){ return 4; //found rolling a die, guaranteed random}

Prerequisite(s): MATH 240.

Course Outline

We will attempt to cover the following topics over the course of the semester:

  1. Discrete Probability Distributions
  2. Continuous Probability Distributions
  3. Combinatorics
  4. Conditional Probability
  5. Distributions and Densities
  6. Expected Value and Variance
  7. Sums of Random Variables
  8. Law of Large Numbers
  9. Central Limit Theorem
  10. Generating Functions
  11. Markov Chains
  12. Random Walks

General Policies

The Pennbook describes general policies every course at Penn. I would particularly draw your attention to the sections on academic integrity (cheating), secular and religious holidays, and students with disabilities. If you need to miss a class (and thus an exercise set) due to observance of a holiday, you must let me know within the first two weeks of school.

Homework Policy

This course will have three types of homework: preparatory, daily, and weekly.

Preparatory. A reading assignment for the material to be covered in the next class will be assigned each day.

Daily. Short exercises will be assigned during each class in order to ensure that students keep up with the course material. These exercises should take no more than an hour to complete and are due at the beginning of the next class unless otherwise specified, and late assignments will not be accepted.

Weekly. Longer problem sets will be assigned weekly — these will require a greater amount of effort than the exercises.

You are encouraged to work with other students on problem sets and exercises, but you must write up the material yourself and acknowledge any assistance you received.

Grading Policy

Daily exercises are due at the beginning of the next class, and no late assignments will be accepted. Problem sets may be turned in late, but the maximum attainable score will drop 20% each day after the due date.

Exercises will be graded mainly on completion and evidence of thought, whereas problem sets will be thoroughly checked.

There will be two in-class exams on 7 October and 11 November. They will each count for 15% of your grade.

Your final numerical grade will be computed as

  • 10% Daily exercises
  • 15% Exam #1
  • 15% Exam #2
  • 30% Problem sets
  • 30% final
At the end of the course, a curve will be fixed and used to assign a final letter grade.

Code

I'll collect the code I mention in class here.

  • Hypergeometric & Benford plots - hb.nb
  • Monte Carlo & Pi — I created an animation of using Monte Carlo to estimate Pi. You can view the animation and download the Mathematica notebook used to create it.
  • Three-card swindle — You can download the Mathematica notebook I showed in class to simulate trials of the three-card swindle.
  • Central Limit Theorem — You can download the Mathematica notebook I showed in class to motivate the central limit theorem for discrete and continuous random variables.

Miscellaneous

File Description
3.2.36.pdf Solution to problem 36 in § 3.2 the book
23septex.pdf Solution to the exercise from 23 Sep

Homework grades and averages for each assignment may be checked here.

Topical

Links pertaining to some of the topics mentioned in class

  • Let's Make a deal applet — Not sure of the Monty Hall solution we talked about? Try it for yourself!
  • I created a Facebook group for the course. Please use this to get in contact with classmates for homework collaborations.
  • An exposition on universality in probability

General

Some online content of possible interest.

Course Text

We will use the free online text Introduction to Probability by Charles M. Grinstead and J. Laurie Snell. If you'd like a paper copy, it is currently on sale from the American Mathematical Society for 29$ + 5$ shipping.

Lecture notes

I will post the notes I base my lectures on here.

Pages 1-10
Pages 11-20
Pages 21-30
Pages 31-40
Pages 41-50
Pages 51-60
Pages 61-70
Pages 71-80

Books on reserve in the library

  1. Richard Brualdi
    Introductory Combinatorics
    Pearson/Prentice Hall
    2004
  2. Chuan Chong Chen
    Principles and Techniques in Combinatorics
    World Scientific
    1992

Download the Random Variable handout. This will not be for a grade, but will help you understand random variables and their distribution/density functions. Please read and think through the questions very carefully.

Exams

Copies of the exams with included solutions are available below.

Exam 1 — due to an unfortunate typo in problem 5, two answers will be accepted.

Exam 2

Daily Exercises

Set Date assigned Date due Exercises
16 16 Nov, 2009 18 Nov, 2009
  1. Grinstead & Snell, §9.1, Exercise 1
  2. Grinstead & Snell, §9.2, Exercise 15
  3. Read "Normal Distributions and genetics", Grinstead & Snell pp. 345-350
15 9 Nov, 2009 13 Nov, 2009 Grinstead & Snell, §8.1, Exercise 9
14 6 Nov, 2009 9 Nov, 2009 Show that convolution is commutative and associative
13 2 Nov, 2009 4 Nov, 2009
  1. Grinstead & Snell, §6.2, Exercise 4
12 30 Oct, 2009 2 Nov, 2009
  1. Flip a coin. If the result was heads, flip twice more; if tails, once more. Let H1 measure the number of heads on the first flip, Ht the total number of heads. Show by explicit computation that E(H1+Ht) = E(H1) + E(Ht).
  2. For a random variable X, Poisson distributed with parameter λ, show that E(X) = λ.
  3. Read §6.1 and complete Exercise 12 of the same section.
11 28 Oct, 2009 30 Oct, 2009
  1. Grinstead & Snell, §6.1, Exercise 1
  2. Grinstead & Snell, §6.1, Exercise 2
10 26 Oct, 2009 28 Oct, 2009
  1. Grinstead & Snell, §5.1, Exercise 35
  2. Show that the probability difference in equation 5.4 of Grinstead & Snell is the probability that a Poisson-distributed variable with parameter (λ t) takes a value n
9 23 Oct, 2009 26 Oct, 2009 Grinstead & Snell, §5.1, Exercise 12
8 12 Oct, 2009 14 Oct, 2009
7 9 Oct, 2009 12 Oct, 2009 Grinstead & Snell, §4.1, Exercise 1
6 30 Sep, 2009 2 Oct, 2009
  1. 10 different paintings are to be distributed to n offices, at most one per office. Find the number of ways to do so when n = 14 and n = 6. Repeat your analysis for the case when the paintings are identical.
  2. Grinstead & Snell, §3.2, Exercise 6
5 23 Sep, 2009 25 Sep, 2009
4 21 Sep, 2009 23 Sep, 2009
  1. Grinstead & Snell, §2.2, Exercise 5
  2. Grinstead & Snell, §3.1, Exercise 4
3 16 Sep, 2009 18 Sep, 2009 Grinstead & Snell, §2.2, Exercise 1
2 14 Sep, 2009 16 Sep, 2009
  2. In roulette, there are 38 slots into which the ball may fall — 18 red, 18 black, and two green. Compute the odds and probability of winning a bet on red.
1 11 Sep, 2009 14 Sep, 2009
  1. Grinstead & Snell, §1.2, Exercise 23
  2. For three tosses of a fair coin, write:
    1. The sample space
    2. The natural (i.e. uniform) distribution function
    3. The probability of tossing two heads in a row

Problem Sets

Set # Date assigned Date due Problems Solutions
8 9 Nov, 2009 20 Nov, 2009 pdf
7 30 Oct, 2009 6 Nov, 2009 pdf pdf, 6.1.10.nb, 6.1.20c.nb
6 23 Oct, 2009 30 Oct, 2009 pdf pdf
5 18 Oct, 2009 23 Oct, 2009 pdf pdf
4 02 Oct, 2009 16 Oct, 2009 pdf pdf / 4.2.10
3 25 Sep, 2009 02 Oct, 2009 pdf
2 18 Sep, 2009 25 Sep, 2009 pdf
1 11 Sep, 2009 18 Sep, 2009 pdf