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Unformatted text preview: 7 Transform Methods 7.1 Introduction 241 7.2 The Characteristic Function 242 7.3 The s-Transform 245 7.4 The z-Transform 250 7.5 Random Sum of Random Variables 256 7.6 Chapter Summary 261 7.7 Problems 261 7.1 Introduction Different types of transforms are used in science and engineering. These include the z-transform, Laplace transform, and Fourier transform. They are sometimes called characteristic functions. One of the reasons for their popularity is that when they are introduced into the solution of many problems, the calculations become greatly simplified. For example, many solutions of equations that involve derivatives and integrals of functions are given as the convolution of two func- tions: a ( x ) * b ( x ) . As students of signals and systems know, the Fourier transform of a convolution is the product of the individual Fourier transforms. That is, if F [ g ( x ) ] is the Fourier transform of the function g ( x ) , then F [ a ( x ) * b ( x ) ] = A ( w ) B ( w ) where A ( w ) is the Fourier transform of a ( x ) and B ( w ) is the Fourier transform of b ( x ) . This means that the convolution operation can be replaced by the much simpler multiplication operation. In fact, sometimes transform methods are the only tools available for solving some types of problems. 241 242 Chapter 7 Transform Methods This chapter discusses how transform methods are used in probability theory. We consider three types of transforms: characteristic functions, the z-transform (or moment-generating function) of PMFs, and the s-transform (or Laplace trans- form) of PDFs. The z-transform and the s-transform are particularly used when random variables take only nonnegative values. Thus, the s-transform is essen- tially the one-sided Laplace transform of a PDF. Examples of this class of ran- dom variables are frequently encountered in many engineering problems, such as the number of customers that arrive at the bank or the time until a component fails. We are interested in the s-transforms of PDFs and z-transforms of PMFs and not those of arbitrary functions. As a result, these transforms satisfy certain conditions that relate to their probabilistic origin. 7.2 The Characteristic Function Let f X ( x ) be the PDF of the continuous random variable X . The characteristic function of X is defined by Phi1 X ( w ) = E bracketleftbig e jwX bracketrightbig = integraldisplay ∞-∞ f X ( x ) e jwx dx where j = √- 1. We can obtain f X ( x ) from Phi1 X ( w ) as follows: f X ( x ) = 1 2 π integraldisplay ∞-∞ Phi1 X ( w ) e- jwx dw If X is a discrete random variable with PMF p X ( x ) , the characteristic function is given by Phi1 X ( w ) = ∞ summationdisplay x =-∞ p X ( x ) e jwx Note that Phi1 X ( ) = 1, which is a test of whether a given function of w is a true characteristic function of a random variable....
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
- Fall '07