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Principles of Communication Prof. V. Venkata Rao 2 CHAPTER 2 U Probability and Random Variables 2.1 Introduction At the start of Sec. 1.1.2, we had indicated that one of the possible ways of classifying the signals is: deterministic or random. By random we mean unpredictable; that is, in the case of a random signal, we cannot with certainty predict its future value, even if the entire past history of the signal is known. If the signal is of the deterministic type, no such uncertainty exists. Consider the signal x ( t ) = A cos ( 2 π f1 t + θ ) . If A , θ and f1 are known, then (we are assuming them to be constants) we know the value of x ( t ) for all t . ( A , θ and f1 can be calculated by observing the signal over a short period of time). Now, assume that x ( t ) is the output of an oscillator with very poor frequency stability and calibration. Though, it was set to produce a sinusoid of frequency f = f1 , frequency actually put out maybe f1' where f1' ∈ ( f1 ± ∆ f1 ) . Even this value may not remain constant and could vary with time. Then, observing the output of such a source over a long period of time would not be of much use in predicting the future values. We say that the source output varies in a random manner. Another example of a random signal is the voltage at the terminals of a receiving antenna of a radio communication scheme. Even if the transmitted 2.1 Indian Institute of Technology Madras

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Principles of Communication Prof. V. Venkata Rao (radio) signal is from a highly stable source, the voltage at the terminals of a receiving antenna varies in an unpredictable fashion. This is because the conditions of propagation of the radio waves are not under our control. But randomness is the essence of communication. Communication theory involves the assumption that the transmitter is connected to a source, whose output, the receiver is not able to predict with certainty. If the students know ahead of time what is the teacher (source + transmitter) is going to say (and what jokes he is going to crack), then there is no need for the students (the receivers) to attend the class! Although less obvious, it is also true that there is no communication problem unless the transmitted signal is disturbed during propagation or reception by unwanted (random) signals, usually termed as noise and interference. (We shall take up the statistical characterization of noise in Chapter 3.) However, quite a few random signals, though their exact behavior is unpredictable, do exhibit statistical regularity. Consider again the reception of radio signals propagating through the atmosphere. Though it would be difficult to know the exact value of the voltage at the terminals of the receiving antenna at any given instant, we do find that the average values of the antenna output over two successive one minute intervals do not differ significantly. If the conditions of propagation do not change very much, it would be true of any two averages (over one minute) even if they are well spaced out in time. Consider even a simpler experiment, namely, that of tossing an unbiased coin (by a person without any magical powers). It is true that we do not know in advance whether the outcome on a particular toss would be a head or tail (otherwise, we stop tossing the coin at the start of a cricket match!). But, we know for sure that in a long sequence of tosses, about half of the outcomes would be heads (If this does not happen, we suspect either the coin or tosser (or both!)). 2.2 Indian Institute of Technology Madras

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Principles of Communication Prof. V. Venkata Rao Statistical regularity of averages is an experimentally verifiable phenomenon in many cases involving random quantities. Hence, we are tempted to develop mathematical tools for the analysis and quantitative characterization of random signals. To be able to analyze random signals, we need to understand random variables. The resulting mathematical topics are: probability theory, random variables and random (stochastic) processes. In this chapter, we shall develop the probabilistic characterization of random variables. In chapter 3, we shall extend these concepts to the characterization of random processes. 2.2 Basics of Probability We shall introduce some of the basic concepts of probability theory by defining some terminology relating to random experiments (i.e., experiments whose outcomes are not predictable). 2.2.1. Terminology Def. 2.1: Outcome The end result of an experiment. For example, if the experiment consists of throwing a die, the outcome would be anyone of the six faces, F1 ,........, F6  Def. 2.2: Random experiment An experiment whose outcomes are not known in advance. (e.g. tossing a coin, throwing a die, measuring the noise voltage at the terminals of a resistor etc.)  Def. 2.3: Random event A random event is an outcome or set of outcomes of a random experiment that share a common attribute. For example, considering the experiment of throwing a die, an event could be the 'face F1 ' or 'even indexed faces' ( F2 , F4 , F6 ). We denote the events by upper case letters such as A , B or A1 , A2 ⋅ ⋅ ⋅ ⋅  2.3 Indian Institute of Technology Madras

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Principles of Communication Prof. V. Venkata Rao Def. 2.4: Sample space The sample space of a random experiment is a mathematical abstraction used to represent all possible outcomes of the experiment. We denote the sample space by S.  Each outcome of the experiment is represented by a point in S and is called a sample point. We use s (with or without a subscript), to denote a sample point. An event on the sample space is represented by an appropriate collection of sample point(s). Def. 2.5: Mutually exclusive (disjoint) events Two events A and B are said to be mutually exclusive if they have no common elements (or outcomes).Hence if A and B are mutually exclusive, they cannot occur together.  Def. 2.6: Union of events The union of two events A and B , denoted A ∪ B , {also written as (A + B ) or ( A or B )} is the set of all outcomes which belong to A or B or both. This concept can be generalized to the union of more than two events.  Def. 2.7: Intersection of events The intersection of two events, A and B , is the set of all outcomes which belong to A as well as B . The intersection of A and B is denoted by ( A ∩ B ) or simply ( A B ) . The intersection of A and B is also referred to as a joint event A and B . This concept can be generalized to the case of intersection of three or more events.  Def. 2.8: Occurrence of an event An event A of a random experiment is said to have occurred if the experiment terminates in an outcome that belongs to A . 2.4 Indian Institute of Technology Madras 

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