## Probability

### Time and Place

Monday, Wednesday and Friday, 3:05pm-3:55pm
Room 235 Physics Building

### Text

Title: Introduction to Probability
Authors: Charles M. Grinstead and J. Laurie Snell
You may also want to buy Probability and Statistics, Third Edition, by Murray Spiegel, John Schiller and R. Alu Srinivasan, in the Schaum's Outlines; it's cheap and has many, many worked out problems.

### Instructor

William K. Allard
029A Physics Building
Phone: 660-2861 E-mail: wka@math.duke.edu
Office Hours: Monday and Friday, 11:30am-2pm and by appointment.

### Homework

A homework assignment will be due about once a week. You may work on this with other people, and are encouraged to do so, provided you acknowledge this when you hand it in.

These may change a bit so always check the link before you start doing the homework. At the top of the homework link you will see the due date. Late homework will not be accepted.

• Homework One Due August 31.
• Homework Two Due September 7.
• Homework Three Due September 14.
• Homework Four Due September 21.
• Homework Five Due October 9..
• Homework Six Due October 12.
• Homework Seven Due October 16.
• Homework Eight Due October 30.
• Homework Nine. Due November 6.
• Homework Ten with Solutions. Due November 30.
• Homework Eleven with Solutions. Due November 30
• Homework Twelve with Solutions.

### Quizzes with solutions

• Quiz One.
• Quiz Two.
• Quiz Three.
• Quiz Four.
• Quiz Five.
• Quiz Six.
• Quiz Seven.
• Quiz Eight.
• Quiz Nine.
• Quiz Ten.
• Quiz Eleven.
• First Makeup Quiz.
• Second Makeup Quiz.

### Tests

There will be two tests and a final exam. The first test will take place on Friday, September 25 and the second test will take place on Friday, November 13.

The average on Test One was 90.1/140=64/100 the standard deviation was 29.72/140=30/100. Here is the answer key.

The average on Test Two 77.55/130=60/100 the standard deviation was 18.91/130=14.5/100. Here is the answer key.

And here, at no extra charge, is a previous final exam with the answers worked out PDF.

And here are some tests from an earlier semester.

• Test One
• Answer key to Test One
• Test Two
• Answer key to Test Two

The quizzes will count 15% of your grade. The tests will each count 20% toward your grade. The final exam will count 35% of your grade. The homework will count 10% of your grade. enough. I try to keep the grade distribution like recent Math 135 grade distributions without giving in too much to inflationary pressures.

### Syllabus

We will cover nearly all of the book. Basically, the homework is the syllabus.

Probability has long been one of the most important branches of applied mathematics. For centuries, because of it use in dealing with with gambling, it has always fascinated individuals who otherwise might not care about anything mathematical. More recently, probability has been crucial in solving engineering problems in such fields as reliability theory and communications. Even more recently, it is being used in biology. It is also essential in formulating and solving some central problems in theoretical physics. Probability is a subdiscipline of pure mathematics as well. In my opinion, it is much more important as a part of applied mathematics.

Given what I've said to this point you will not be surprised to find that the course will be oriented toward applications. It is not, however a "cookbook" course in that you will find that you cannot solve the problems without first establishing some conceptual framework for the problem. This and the necessity of using several variable calculus when dealing with continuous random variables do cause some students problems. In order to avoid these problems the student must think on occasion about concepts and might find it necessary to review several variable calculus, particularly multiple integration.

The course will be based on the problems. In particular, if you can master the assigned problems you should do well on the tests. The best way to study for the tests will be to review the problems.

I enjoy talking to students outside of class. Don't hesitate to ask me questions. I am intense but very friendly. If there is something you think I should do another way I hope you will tell me immediately. I may not agree with you but I will hear you out.

I have been accused of talking over the heads of students on occasion. Often the accusers have been right. That is because, like most mathematicians, I am obsessed with the internal consistency and elegance of what I am doing in terms that are clear to me at the expense of communication. The way to prevent this is to speak up. So if you don't understand what I am doing, squawk; it is very likely that many others are lost, too. Of course it helps if you come part of the way and learn the language I use. I believe it is an extremely good one.

### Class notes

These will become more important toward the end of the semester.

These may change a bit so always check the link before you start doing the homework. At the top of the homework link you will see the due date. Late homework will not be accepted.

• A simple finite probability space
• Problem 3.1(18)
• Bayes' Theorem
• The negative binomial distribution
• The expectation of a continuous random variable
• The strong law of large numbers
• The Fourier inversion formula
• The central limit theorem
• Polling
• Confidence intervals (goes with polling above)
• Sums of independent normals
• Conditional expectation
• A summary of lots of stuff
• Transformations of continuous random vectors
• Gaussian random vectors

### Other Goodies

Here are links to materials which can either replace or supplement the text. They will be revised from time to time so note the revision dates.

### Combinatorial Analysis.

• Basic counting PDF
• How many ways can you put m things in n boxes? PDF
• ### Axioms of Probability.

• Probability spaces PDF
• A simple finite probability space PDF
• Relative frequency PDF
• ### Conditional Probability and Independence.

• A good example PDF
• Things in a row PDF
• Finding your keys in the dark PDF
• An example of Bayes' rule PDF
• Let's make a deal and conditional probability PDF
• Some worked conditional probability problems PDF
• Computing expectation by conditioning PDF
• Draw until only white are left PDF
• Two sevens before six evens PDF
• A very useful formula PDF
• ### Random Variables.

• Some basic random variable stuff PDF
• The binomial distribution PDF
• The cumulative distribution function PDF
• Expectation PDF
• More on expectation PDF
• The probability generating function PDF
• Computing expectation and variance in a simple example. PDF
• Some worked homework problems. PDF
• The Poisson process, Part One PDF
• The Poisson process, Part Two PDF
• ### Continuous Random Variables.

• Continous random variables PDF
• The density of a function of a continuous random vector PDF
• Computing the density of the product of two independent uniformly distributed random variables PDF
• Normal random variables PDF
• A table of the Phi function HTM
• Some worked problems from Chapter Five PDF
• ### Jointly Distributed Random Variables.

• Random vectors PDF
• More and better stuff on random vectors PDF
• Convolutions and Fourier transforms PDF
• Transformations of continuous random vectors PDF
• The multinomial distribution PDF
• Some basic facts about integration PDF
• Some worked problems from Chapter Six PDF
• Conditional distributions; the continuous case PDF
• Problem 58 on page 295 PDF
• ### Properties of Expectation.

• Conditional density PDF
• What's at the heart of n. 25 on p. 383 PDF
• ### Limit Theorems.

• The central limit theorem PDF
• A sampling theorem PDF
• Gaussian random vectors PDF
• ### Other stuff

• Random walk PDF