This is a second course in probability theory, with emphasis on the theory of random processes. It assumes a good knowledge of Mathematics 5651 (Introduction to Probability Theory) or a probability course at a similar level, as well as knowledge of linear algebra at the level of second-year calculus. This course can be counted toward a math major's analysis requirement as described here . It can be used to fulfill one of the requirements for mathematics majors who are pursuing the actuarial specialization, and is a suggested course for the mathematics education specialization.
The focus of Math 5652 is on Markov chains in discrete and continuous time. Markov chains are families of random variables, usually representing the random behavior of a system as it evolves in time. The structure of a Markov chain is simple enough to be analyzed in detail, but complicated enough to apply to many interesting situations. Markov chains are used as models for a wide class of phenomena in statistics, physics, biology, medicine, finance, and other fields.
Precise definitions and statements of theorems are important in this course, with some proofs required.
To suggest some flavor of Math 5652, a student might learn how to ...
A sample problem related to continuous-time Markov chains:
Consider a two-station queueing network in which arrivals only occur at the first server and do so at rate 2 per unit time. If a customer finds server 1 free the customer enters the system; otherwise the customer goes away. When a customer is done at the first server she moves on to the second server if it is free and leaves the system if it is not. Suppose that server 1 serves at rate 4 while server 2 serves at rate 2. Formulate a Markov chain model for this system with state space {0, 1, 2, 1_2}, where the state indicates the servers which are busy.
- In the long run, what proportion of customers actually enter the system?
- In the long run, what proportion of customers visit server 2?