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.

TOPICS DISCUSSED:

I. INTRODUCTION

II. PROBABILITY THEORY

III. RANDOM VARIABLES

IV. FUNCTIONS OF A RANDOM VARIABLE

V. TWO AND MORE RANDOM VARIABLES

VI. SEQUENCES OF RANDOM VARIABLE

VII. STOCHASTIC PROCESSES: INTRODUCTION

VIII. SECOND-ORDER PROCESSES

IX. APPLICATIONS OF RANDOM PROCESSES