Contact information
Office	DRL 3C7
Office hours	By appointment, or just stop by
Phone	86219
Email	guf...@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

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.

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

Discrete Probability Distributions
Continuous Probability Distributions
Combinatorics
Conditional Probability
Distributions and Densities
Expected Value and Variance
Sums of Random Variables
Law of Large Numbers
Central Limit Theorem
Generating Functions
Markov Chains
Random Walks

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.

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.

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.

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.


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.

Links pertaining to some of the topics mentioned in class

An explanation of Google's PageRank algorithm. See also the brilliantly named $25 Billion Eigenvector.
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

Some online content of possible interest.

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.

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

Richard Brualdi

Introductory Combinatorics

Pearson/Prentice Hall

2004
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.

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

Final Exam


Set	Date assigned	Date due	Exercises

20	2 Dec, 2009	4 Dec, 2009	Grinstead & Snell, §11.1, Exercise 5 Grinstead & Snell, §11.1, Exercise 11
19	2 Dec, 2009	30 Nov, 2009	Grinstead & Snell, §10.1, Exercise 1
18	23 Nov, 2009	25 Nov, 2009	Let X:Ω→{1,2,…,n} be a random variable with uniform distribution. Given that find the expected value and variance by computing moments from g(t).
17	20 Nov, 2009	23 Nov, 2009	Grinstead & Snell, §9.2, Exercise 4
16	16 Nov, 2009	18 Nov, 2009	Grinstead & Snell, §9.1, Exercise 1 Grinstead & Snell, §9.2, Exercise 15 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	Grinstead & Snell, §6.2, Exercise 4
12	30 Oct, 2009	2 Nov, 2009	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). For a random variable X, Poisson distributed with parameter λ, show that E(X) = λ. Read §6.1 and complete Exercise 12 of the same section.
11	28 Oct, 2009	30 Oct, 2009	Grinstead & Snell, §6.1, Exercise 1 Grinstead & Snell, §6.1, Exercise 2
10	26 Oct, 2009	28 Oct, 2009	Grinstead & Snell, §5.1, Exercise 35 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	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. Grinstead & Snell, §3.2, Exercise 6
5	23 Sep, 2009	25 Sep, 2009
4	21 Sep, 2009	23 Sep, 2009	Grinstead & Snell, §2.2, Exercise 5 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	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	Grinstead & Snell, §1.2, Exercise 23 For three tosses of a fair coin, write: The sample space The natural (i.e. uniform) distribution function The probability of tossing two heads in a row


Set #	Date assigned	Date due	Problems	Solutions

9	29 Nov, 2009	4 Dec, 2009	pdf	pdf
8	9 Nov, 2009	20 Nov, 2009	pdf	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

Fall 2009

430 Calendar

Wednesday, September 9

Time:	1:00 PM - 2:00 PM
Summary:	Discrete Probability Distributions

Friday, September 11

Time:	1:00 PM - 2:00 PM
Summary:	Discrete Probability Distributions

Monday, September 14

Time:	1:00 PM - 2:00 PM
Summary:	Discrete Probability Distributions

Wednesday, September 16

Time:	1:00 PM - 2:00 PM
Summary:	Continuous Probability Distributions

Friday, September 18

Time:	1:00 PM - 2:00 PM
Summary:	Continuous Probability Distributions

Monday, September 21

Time:	1:00 PM - 2:00 PM
Summary:	Continuous Probability Distributions

Wednesday, September 23

Time:	1:00 PM - 2:00 PM
Summary:	Combinatorics

Friday, September 25

Time:	1:00 PM - 2:00 PM
Summary:	Combinatorics

Monday, September 28

Time:	1:00 PM - 2:00 PM
Summary:	Combinatorics

Wednesday, September 30

Time:	1:00 PM - 2:00 PM
Summary:	Combinatorics

Friday, October 2

Time:	1:00 PM - 2:00 PM
Summary:	Combinatorics

Monday, October 5

Time:	1:00 PM - 2:00 PM
Summary:	Review

Wednesday, October 7

Time:	1:00 PM - 2:00 PM
Summary:	Exam 1

Friday, October 9

Time:	1:00 PM - 2:00 PM
Summary:	Random Variables

Monday, October 12

Time:	1:00 PM - 2:00 PM
Summary:	Conditional Probability

Wednesday, October 14

Time:	1:00 PM - 2:00 PM
Summary:	Conditional Probability

Friday, October 16

Time:	1:00 PM - 2:00 PM
Summary:	Conditional Probability

Monday, October 19

Time:	1:00 PM - 2:00 PM
Summary:	No Class (Fall Break)

Wednesday, October 21

Time:	1:00 PM - 2:00 PM
Summary:	Conditional Probability

Friday, October 23

Time:	1:00 PM - 2:00 PM
Summary:	Distributions and Densities

Monday, October 26

Time:	1:00 PM - 2:00 PM
Summary:	Distributions and Densities

Wednesday, October 28

Time:	1:00 PM - 2:00 PM
Summary:	Expected Value and Variance

Friday, October 30

Time:	1:00 PM - 2:00 PM
Summary:	Expected Value and Variance

Monday, November 2

Time:	1:00 PM - 2:00 PM
Summary:	Expected Value and Variance

Wednesday, November 4

Time:	1:00 PM - 2:00 PM
Summary:	Sums of Random Variables

Friday, November 6

Time:	1:00 PM - 2:00 PM
Summary:	Sums of Random Variables

Monday, November 9

Time:	1:00 PM - 2:00 PM
Summary:	Law of Large Numbers

Wednesday, November 11

Time:	1:00 PM - 2:00 PM
Summary:	Exam 2
Description:	This exam will concentrate on Conditional Probability, Important Distributions and Densities, and Expected Value and Variance

Friday, November 13

Time:	1:00 PM - 2:00 PM
Summary:	Law of Large Numbers

Monday, November 16

Time:	1:00 PM - 2:00 PM
Summary:	Central Limit Theorem

Wednesday, November 18

Time:	1:00 PM - 2:00 PM
Summary:	Central Limit Theorem

Friday, November 20

Time:	1:00 PM - 2:00 PM
Summary:	Central Limit Theorem

Monday, November 23

Time:	1:00 PM - 2:00 PM
Summary:	Generating Functions

Wednesday, November 25

Time:	1:00 PM - 2:00 PM
Summary:	Generating Functions

Friday, November 27

Time:	1:00 PM - 2:00 PM
Summary:	No Class (Thanksgiving)

Monday, November 30

Time:	1:00 PM - 2:00 PM
Summary:	Generating Functions

Wednesday, December 2

Time:	1:00 PM - 2:00 PM
Summary:	Markov Chains

Friday, December 4

Time:	1:00 PM - 2:00 PM
Summary:	Markov Chains

Monday, December 7

Time:	1:00 PM - 2:00 PM
Summary:	Markov Chains

Wednesday, December 9

Time:	1:00 PM - 2:00 PM
Summary:	Markov Chains

Josh Guffin - Teaching

Math 430, Introduction to Probability

Fall 2009

Fall 2009