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EE 560 deals with the mathematical theory that was developed
in the last century to describe and analyze the variety of "random"
phenomena that occur in the various branches of electrical engineering
and other fields. This theory developed after a long period during
which "chance phenomena" had been dealt with using a variety of ad hoc
approaches, many of which were inconsistent and lead to mathematicians
disparaging the whole field claiming that "probability theory" was not
a proper mathematical subject. This situation was altered by a Soviet
school of mathematicians led by A.N. Kolmogorov, which, in the 1930's,
finally established a proper mathematical basis for a theory of
probability. The theory of probability is intended to reflect a
particular view we sometimes adopt to explain how the world behaves.
Underlying this theory are two basic notions: the notion of the outcome
of an experiment being random, and the concept of the probability of a
particular type of outcome. Recent physics has seen the evolution of a
nondeterministic theory of quantum mechanics, which has seemed able to
accurately predict many observed phenomena the deterministic theories
could not. Notwithstanding the success of this nondeterministic theory
in applications, not everyone is convinced of the fundamental validity
of random models for our world (e.g., we have Einstein's often quoted
view in reference to his reluctance to embrace quantum mechanics as a
fundamental description of reality, that "God does not play dice" ...
to which Niels Bohr’s retort “Who are you to tell God what to
do”). The debate on the fundamental nature of the universe is a
metaphysical debate - seemingly without resolution on which we will not
dwell. Instead, we shall simply agree to accept this model as our
(ideal) concept of what we refer to as a random event and set aside the
question of whether or not it is as an absolutely accurate description
of reality. A justification of random models in practice is
possible on pragmatic grounds quite apart from accepting the existence
of "ideally random" phenomena. In the real (causal) world, we are often
faced with events which we might regard as deterministic, but for which
we do not know some or all of the determining factors, e.g., weather
prediction or stock market fluctuations. In such cases, we frequently
model the event as random although we fully accept that it would not be
so according to our ideal notion above. Therefore, we admit ignorance
and use probabilistic tools to measure likelihood of our
predictions. Before we begin the mathematical discussion, we
examine these concepts to understand how the mathematical theory is
supposed to relate to the physical world and application.
The text used in this course is a classic one. It is neither a
handbook, nor a tutorial book. It contains a logical sequence of topics
presented in a smooth and continuous manner. |