3-1 The s Transform Let fz(zo) .be any PDF. The exponential transform (or s transform) We are interested only in the s transforms of PDF's and not of arbitrary functions. Thus, we need note only those aspects of transform theory which are relevant to this special case. As long as fx(xo) is a PDF, the above integral must be finite at least for the case where s is a pure imaginary (see Prob. 3.01). Furthermore, it can be proved that the s transform of a PDF is unique to that PDF. Three examples of the calculation of s transforms follow: First, consider the PDF [The unit step function p-l(xO -a) is defined in Sec. 2-9.1 For a second example, we consider the uniform PDF ifO_<xo_<l = p-1(20 -0) -p-l(x0 -1)fz(xO)= { 0 1 otherwise Our third example establishes a result to be used later. Consider the PDF for a degenerate (deterministic) random variable x which always takes on the -experimental value a, The PDF corresponding to a given s transform, fZT(s), is known as the inverse transform of fxT(s). The formal technique for obtaining inverse transforms is beyond the scope of the mathematical prerequisites assumed for this text. For our purposes, we shall often be able to THE 2 TRANSFORM 99 obtain inverse transforms by recognition and by exploiting a few simple properties of transforms. A discussion of one simple procedure for attempting to evaluate inverse transforms mill appear in our solution to the example of Sec. 3-8. 3-2 The z Transform Once we are familiar with the impulse function, any PMF may be expressed as a PDF. To relate the PMF pZ(xo) to its corresponding PDF fz(xo), we use the relation f&o) = 2 pZ(a)po(xo-a) a As an example, the PAIF pz(xo) shown below I 0.50 SO-1 H) otherwise may be written as the PDF f,(xo), The s transform of this PDF is obtained from where we have made use of the following relation from Sec. 2-9 The above s transform could also have been obtained directly from the equivalent (expected value) definition of fzT(s),- - - - - 100 TRANSFORMS AND SOME APPLICATIONS Alt,hough the s transform is defined for the PDF of any random variable, it is convenient to define one additional type of transform for a certain type of PMF. If pz(xo)is the PMF for a discrete random variable which can take on only nonnegative integer experimental values (xo = 0, 1, 2, . . .), we define the disc~xeletransform (or z transform) of pz(xo)to be pZT(z),given by - - - m-----w pzT(z) --E(zz) = 2 zspz(xo) = -=zo =O = = We do riot find it particularly useful to define a z transform for PAIF's which allow ioni integer or negative experimental values. In practice, a large number of discrete random variables arise from a count of integer units and from the quantization of a positive quantity, and it is for cases like these that our nonnegative integer con~t~raintholds. The PlIF at the start of this section allows only nonnegative integer values of its random variable. As an example, we obtain the z transform of this PSIF, Xote that the x transfornl for a PJIF may be obtained from the s transform of the equivalent I'D17 by substituting z = e-". The z transform earl be shown to be finite for at least lzl 5 1 and to be unique to its 1'3IE'. We shsli normally go back to a 1'AII' from its transfornl by rec~og~iit.iorrof a few familiar tra.nsforms. How-ever, we can note from the definition of pZT(z), that it is possible to determine the individual ternxi of the PMF from 3-3 Moment-generating Properties of the Transforms Consider the nth derivative with respect to s of fzT(s), MOMENT-GENERATING PROPERTIES OF THE TRANSFORMS I01 The right-hand side of the last equation, when evaluated at s = 0, may be recognized to be equal to (-l)"E(xn). Thus, once we obtain the s transform for a I'DF, we can find all the moments by repeated differentiation rather than by performing other integrations. From the above expression for the nth derivative of fzT(s),we may establish the following useful results: Of course, when certain moments of a PDF do not exist, the correspond-ing derivatives of fzT(s)will be infinite when evaluated at s = 0. As one example of the use of these relations, consider the PDF ..J;(xo) = p-l(xo -O)Xe-hzo,for which we obtained fZT(s)= X/(s + A) in Sec. 3-1. We may obtain E(x) and uZ2by use of the relations The moments for a 1'31 F may also be obtained by differentiation of its z transform, although the resulting equations are somewhat dif-ferent from those obtairied above. Begirining with the definition of the z transform, we have In general, for n = 1, 2, . . . , we have---- 102 TRANSFORMS AND SOME APPLICATIONS The right-hand side of this last equation, when evaluated at z = 1, is equal to some linear conlbination of E(xn), E(xn-I), . . . , E(x2), and B(x). What we are accomplishing here is the determination of all moments of a PAIF from a single summation (the calculation of the transform itself) rather than attempting to perform a separate sum- mation directly for each moment. This saves quite a bit of work for those PAIF'S whose z transforms may be obtained in closed form. We shall frequently use the following relations which are easily obtained from the above equations: -We often recognize sums which arise in our work to be similar to expressions for moments of PMF's, and then we may use z trans-forms to carry out the summations. (Examples of this procedure arise, for instance, in the solutions to Probs. 3.10 and 3.12.) As an example of the moment-generating properties of the z transform, consider the geometric PNF defined by 3-4 A similar relation which will be used frequently in our work with z transforms is Sums of Independent Random Variables; Convolution The properties of sums of independent random variables is an impor- tant topic in the study of probability theory. In this section we approach this topic from a sample-space point of view. A transform approach will be considered in Sec. 3-5, and, in Sec. 3-7, we extend our work to a matter involving the sum of a random number of random variables. Sums of independent random variables also will be our main concern when we discuss limit theorems in Chap. 6. To begin, we wish to work in an xo,y0 event space, using the method of Sec. 2-14, to derive the PDF for w, the sum of two random variables x -and y. After a brief look at the general ease, we shall specialize
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