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

ISBN-10: 0470544147

ISBN-13: 9780470544143

ISBN-10: 0879422351

ISBN-13: 9780879422356

This "bible" of an entire iteration of communications engineers used to be initially released in 1958. the focal point is at the statistical concept underlying the learn of signs and noises in communications platforms, emphasizing concepts in addition s effects. finish of bankruptcy difficulties are provided.Sponsored by:IEEE Communications Society

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**Example text**

4-2) refers to the sample space of y. The simplest application of Eq. (4-1) is that in which g(x) = z. E(x) is usually called the mean value of x and denoted by m:/e. Now suppose x is a continuous random variable with the probability density Pl(X) and g(x) is a single-valued function of x. Again we want to determine the average of the random variable g(x). Let x be approximated by a discrete random variable x' which takes on the values X m with probability Pl(Xm) ~Xm, where the M intervals dXm partition the sample space of x.

Again we want to determine the average of the random variable g(x). Let x be approximated by a discrete random variable x' which takes on the values X m with probability Pl(Xm) ~Xm, where the M intervals dXm partition the sample space of x. Then by Eq. ) ax.. ",-1 If we let all the Ax", --+ 0, thus forcing M ~ 00, the limiting value of this sum is the integral given below in Eq. (4-3), at least for piecewise continuous g(x) and Pl(X). This procedure suggests that we define the statistical average of the continuous random variable g(x) by the equation E[g(x)] = J-+..

Henceforth we will use, therefore, the probability density function, if convenient, whether we are concerned with continuous, discrete, or mixed random variables. Joint Probability Density Functions. In the case of a single random variable, the probability density function was defined as the derivative of the probability distribution function. Similarly, in the two-dimensional case, if the joint probability distribution function is everywhere continuous and possesses a continuous mixed second partial derivative everywhere except possibly on a finite set of curves, we may define the joint probability density function as this second derivative a2 ax aY P(x p(X,Y) = Then s X,y s P(x Y) ~ X,y s (3-22) Y) = J~.

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

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