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prob_random_proc_mar11

# prob_random_proc_mar11 - Chapter 5 Probability and Random...

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Chapter 5 Probability and Random Processes From Introduction to Communication Systems Copyright by Upamanyu Madhow, 2008-2011 Probability theory is fundamental to communication system design, especially for digital commu- nication. Not only are there uncontrolled sources of uncertainty such as noise, interference, and other channel impairments that are only amenable to statistical modeling, but the very notion of information underlying digital communication is based on uncertainty. In particular, the receiver in a communication system does not know a priori what the transmitter is sending (otherwise the transmission would be pointless), hence the receiver designer must employ statistical models for the transmitted signal. In this chapter, we review basic concepts of probability and random variables with examples motivated by communications applications. We also introduce the con- cept of random processes, which are used to model both signals and noise in communication systems. 5.1 Probability Basics In this section, we remind ourselves of some important definitions and properties. Sample Space: The starting point in probability is the notion of an experiment whose outcome is not deterministic. The set of all possible outcomes from the experiment is termed the sample space Ω. For example, the sample space corresponding to the throwing of a six-sided die is Ω = { 1 , 2 , 3 , 4 , 5 , 6 } . An analogous example which is well-suited to our purpose is the sequence of bits sent by the transmitter in a digital communication system, modeled probabilistically by the receiver. For example, suppose that the transmitter can send a sequence of seven bits, each taking the value 0 or 1. Then our sample space consists of the 2 7 = 128 possible bit sequences. Event: Events are sets of possible outcomes to which we can assign a probability. That is, an event is a subset of the sample space. For example, for a six-sided die, the event { 1 , 3 , 5 } is the set of odd-numbered outcomes. We are often interested in probabilities of events obtained from other events by basic set opera- tions such as complementation, unions and intersections; see Figure 5.1. Complement of an Event (“NOT”): For an event A , the complement (“not A ”), denoted by A c , is the set of outcomes that do not belong to A . 1

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c B A Union A B Intersection Complement A Complement Sample Space A Figure 5.1: Basic set operations. Union of Events (“OR”): The union of two events A and B , denoted by A B , is the set of all outcomes that belong to either A or B . The term ”or” always refers to the inclusive or , unless we specify otherwise. Thus, outcomes belonging to both events are included in the union. Intersection of Events (“AND”): The intersection of two events A and B , denoted by A B , is the set of all outcomes that belong to both A and B .
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