## Mathematics 571 Introduction to Probability Models, Spring 2012

### Instructor: Eric Key

Calculus and differential equations are used to construct models where there is no uncertainty. In this course, we will incorporate uncertainty into models of systems. For example, we can consider the decay of a radioactive substance as occurring because at random times, individual atoms decay. Instead of asking for the mass of the substance at a particular time, we can ask for the probability distribution of the mass at that time. Other examples include the value of the Standard and Poors 500 stock index, the number of people in line at a grocery check-out, the number of descendents of a in a branch of a family tree in a given generation, and the distribution of gas particles in a box at a given time.

The primary text book for the course is Introduction to Probability Models, Tenth Edition by Sheldon Ross. We will assume that participants have a working knowledge of the probability and statistics at the level of the first three chapters of this text. We will begin with an overview of stochastic processes before beginning with the fourth chapter of the text, Markov Chains. If need be, we can always return to these earlier chapters to fill in any gaps in background. The aim of the text and of the course is very applied. Our goal is to show how simple probability tools can be used to describe a large number of systems with two common characteristics: the systems evolve with time, and uncertainty is a primary consideration.

Some of the things we will investigate are

• Compound Poisson Processes and Insurance Claims.
• The Gambler's ruin problem.
• Random walks
• Branching processes.
• Queuing systems.
• Pricing stock options.

To be prepared for this course you must have some background in elementary probability, calculus, linear differential equations, and matrix algebra. The ability to write simple computer programs or to use MAPLE will be helpful, but is not essential.

If you have any doubts about your preparation, or any other questions, contact me .

There will be (mostly) weekly homework which will account for 50% of your grade, a midterm exam that is 20% of your grade (probably right before spring break) and a final exam (see the final exam schedule for date and time) which will account for 30% of your grade. This is a U/G course, so graduate students will be expected to solve additional and/or more challenging problems.

I will email you your reading assignments, additional lecture notes and homework assignments as the semester goes along.