By William L. Root Jr.; Wilbur B. Davenport

ISBN-10: 0470544147

ISBN-13: 9780470544143

ISBN-10: 0879422351

ISBN-13: 9780879422356

This "bible" of a complete iteration of communications engineers used to be initially released in 1958. the focal point is at the statistical idea underlying the research of indications and noises in communications structures, emphasizing strategies in addition s effects. finish of bankruptcy difficulties are provided.Sponsored by:IEEE Communications Society

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**Additional info for An Introduction to the Theory of Random Signals and Noise**

**Example text**

10) and Dooh (II, Chap. I, Art. 11). : See, for example, Courant (I, Vol. II, Chap. 4, Appendix Art. 5) or Guillemin (II, Ohap. VII, Artl. 19). 51 AVlCBAGES If both sides of this equation are evaluated at v == 0, the integral becomes the first moment of the random variable x. Hence, on solving for that moment, we get (4-20) and we see that the first moment may be obtained by differentiation of the characteristic function. (jIJ) dvn = j" f+" x" exp(jIJx)p(x) dx _ GO Evaluating at tJ = 0, the integral becomes the nth moment of the random variable x.

Note that 36 RANDOM SIGNALS AND NOISE the number of random variables y. I. Suppose that a random variable, is defined as the sum of the real random variables x and 1/: 2 == g(x,y) == Z + 11 (3-48) and that the joint probability functions for z and 'Yare known. ) of the random variable e. Since z and 11 are real random variables, their joint sample space may be taken as the (x,y) plane consisting of the points (- 00 :S z S + 00, - 00 ~ y S + 00). Also,. since z and 'Yare real, 2 must be real, too, and its sample space may be taken as the real line (- 00 :S z ~ too).

Term "random variable" for a fUDction is dictated by tradition. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS P(x S - co) = 0 and P{~ S +00) == 1 21 (3-1) It further follows that the probability that the random variable x falls in the interval a < x ~ b is simply the difference between the values of the probability distribution function obtained at the endpoints of the interval: P(x 5 b) - P(x s a) = P(a < x 5 b) ~ 0 (3-2) The right-hand inequality follows from the nonnegativeness of probabilities.

### An Introduction to the Theory of Random Signals and Noise by William L. Root Jr.; Wilbur B. Davenport

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