Back to the teaching page._______________
Instructor: Professor Vladimir Rotar; Office GMCS -514, phone: 594 7244; e-mail: rotar@sciences.sdsu.edu or vrotar@euclid.ucsd.edu.
Classes:
MW,
Office hours:
MW,
Text: Howard Taylor and Samuel Karlin (1998). An Introduction to Stochastic Modeling, 3rd Edition Academic Press.
Other References:
Examinations. There will be many
quizzes, and a final exam. Homework will be assigned almost each class and will
be collected and graded, as a rule, each week.)
This course may be viewed as a continuation of the
introductory probability course. The main step we take here consists in
considering random phenomena in their dynamics. Not just the number of people
who will live in San Diego in the next year (which is a random variable) but
the process of the San Diego population growth during a certain period (and a
tendency of this growth); not just the tomorrow stock prices but their
evolution in the long run; not just the number of customers a company will deal
with tomorrow but the flow of customers during a particular season; etc.,
etc.
What do we study from a mathematical
point of view? The graph of, say, the air temperature against time during the
whole day is a curve; if this day is tomorrow, it is a random curve, or a
random process. So, while the object of study in the first course on
Probability is random variables, the object of study in Random Processes course
is random functions.
Three main questions
arise.
·
How or in what terms should we describe random processes (or
random functions, which is the same), what new notions should we
introduce?
·
What kinds of processes are most important, how to classify them?
·
What qualitatively new phenomena can we watch when we switch from
the static to the dynamic framework? What qualitatively new and interesting
facts can we discover?
A good example of such facts is ergodicity. For example, the tomorrow temperature strongly
depends on the today temperature, the dependence of the today temperature and
that in a week is much weaker, while the temperature, say, in three months
practically does not depend on what temperature we have today. Thus the system
“is forgetting the past”, the influence of the starting state is vanishing in
time. When it is the case, under what conditions we can watch such phenomena is
one of the main questions of the stochastic processes theory.
The prerequisite is ordinary Calculus and Linear Algebra (not complicated, but
the student should know how to
differentiate and integrate, how to multiply matrices, and what their
eigenvalues are), and an introductory course on Probability Theory (say, Stat-550
or Stat-551a are enough).