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Chapter 10. A Taste of Probability Theory

Chapter 10. A Taste of Probability Theory - 10 A Taste of...

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0-8493-????-?/00/$0.00+$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC 10 A Taste of Probability Theory* 10.1 Introduction In addition to its everyday use in all aspects of our public, personal, and lei- sure lives, probability plays an important role in electrical engineering prac- tice in at least three important aspects. It is the mathematical tool to deal with three broad areas: 1. The problems associated with the inherent uncertainty in the input of certain systems. The random arrival time of certain inputs to a system cannot be predetermined; for example, the log-on and the log-off times of terminals and workstations connected to a com- puter network, or the data packets’ arrival time to a computer network node. 2. The problems associated with the distortion of a signal due to noise. The effects of noise have to be dealt with satisfactorily at each stage of a communication system from the generation, to the transmission, to the detection phases. The source of this noise may be due to either Fuctuations inherent in the physics of the problem (e.g., quantum effects and thermal effects) or due to random distortions due to externally generated uncontrollable parameters (e.g., weather, geography, etc.). 3. The problems associated with inherent human and computing machine limitations while solving very complex systems. Individual treatment of the dynamics of very large number of molecules in a material, in which more than 10 22 molecules may exist in a quart-size con- tainer, is not possible at this time, and we have to rely on statistical averages when describing the behavior of such systems. This is the ±eld of statistical physics and thermodynamics. ²urthermore, probability theory provides the necessary mathematical tools for error analysis in all experimental sciences. It permits estimation of the
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© 2001 by CRC Press LLC error bars and the confdence level For any experimentally obtained result, through a methodical analysis and reduction oF the raw data. In Future courses in probability, random variables, stochastic processes (which is random variables theory with time as a parameter), inFormation theory, and statistical physics, you will study techniques and solutions to the diFFerent types oF problems From the above list. In this very brieF introduction to the subject, we introduce only the very Fundamental ideas and results — where more advanced courses seem to almost always start. 10.2 Basics Probability theory is best developed mathematically based on a set oF axioms From which a well-defned deductive theory can be constructed. This is reFerred to as the axiomatic approach. We concentrate, in this section, on developing the basics oF probability theory, using a physical description oF the underlying concepts oF probability and related simple examples, to lead us intuitively to what is usually the starting point oF the set theoretic axiom- atic approach.
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Chapter 10. A Taste of Probability Theory - 10 A Taste of...

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