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APPENDIX H INTRODUCTION TO PROBABILITY AND RANDOM PROCESSES This appendix is not intended to be a definitive dissertation on the subject of random processes. The major concepts, definitions, and results which are employed in the text are stated here with little discussion and no proof. The reader who requires a more complete presentation of this material is referred to any one of several excellent books on the subject: among them Davenport and Root (Ref. 2), Laning and Battin (Ref. 3), and Lee (Ref. 4). Possibly the most important function served by this appendix is the definition of the notation and of certain conventions used in the text. PROBABILITY Consider an event E which is a possible outcome of a random experiment. We denote by P(E) the probability of this event, and think of it intuitively as the limit, as the number of trials becomes large, of the ratio of the number of times E occurred to the number of times the experiment was tried. The joint event that A and B and C, etc., occurred is denoted by ABC . , and the probability of this joint event, by P(ABC. . ). If these events A, B, C, etc., are mutually independent, which means that the occurrence of any one of them bears no relation to the occurrence of any other, the probability of the joint event is the product of the probabilities of the simple events. That is, P(ABC. . . ) = P(A)P(B)P(C) . . (H- 1)
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630 PROBABILITY AND RANDOM PROCESSES if the events A, B, C, etc., are mutually independent. Actually, the mathe- matical definition of independence is the reverse of this statement, but the result of consequence is that independence of events and the multiplicative property of probabilities go together. RANDOM VARIABLES A random variable Xis in simplest terms a variable which takes on values at random; it may be thought of as a function of the outcomes of some random experiment. The manner of specifying the probability with which different values are taken by the random variable is by the probability distribution function F(x), which is defined by or by the probability density function f(x), which is defined by The inverse of the defining relation for the probability density function is (H-4) An evident characteristic of any probability distribution or density function is From the definition, the interpretation off (x) as the density of probability of the event that X takes a value in the vicinity of x is clear. F(x + dx) - F(x) f(x) = lim &+o dx P(x < X s x + dx) = lim dx40 dx This function is finite if the probability that X takes a value in the infinitesimal interval between x and x + dx (the interval closed on the right) is an infini- tesimal of order dx. This is usually true of random variables which take values over a continuous range. If, however, X takes a set of discrete values xi with nonzero probabilities pi, f (x) is infinite at these values of x. This is accommodated by a set of delta functions weighted by the appropriate probabilities.
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