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Unformatted text preview: 7.6 Chapter Summary 261 Since the number of items purchased by each customer is independent of the number purchased by other customers, we have that G Y  K ( z ) = [ G N ( z ) ] k Thus, G Y ( z ) = summationdisplay k = p K ( k ) G Y  K ( z ) = summationdisplay k = p K ( k ) [ G N ( z ) ] k = G K ( G N ( z )) Since G K ( z ) = e ( z 1 ) and G N ( z ) = e ( z 1 ) , we have that G Y ( z ) = e ( e ( z 1 ) 1 ) trianglesolid 7.6 Chapter Summary This chapter discussed three transform methods that are frequently used in the analysis of probabilistic problems. These are the characteristic function, the stransform, and the ztransform. Both the stransform and the ztransform are used for random variables that take only nonnegative values, which include many random variables that are used to model practical systems. The moment generating properties of the different transforms have also been demonstrated. Table 7.1 is a summary of the different transforms of some of the wellknown PMFs and PDFs. 7.7 Problems Section 7.2: Characteristic Functions 7.1 Find the characteristic function of the random variable X with the following PDF: f X ( x ) = braceleftBigg 1 b a a < x < b otherwise 7.2 Find the characteristic function of the random variable Y with the following PDF: f Y ( y ) = braceleftbigg 3 e 3 y y otherwise 262 Chapter 7 Transform Methods Table 7.1 Summary of the Transforms of WellKnown PMFs and PDFs Characteristic PMF PDF zTransform sTransform Function Bernoulli p X ( x ) = braceleftbigg 1 p x = p x = 1 1 p + zp Binomial p X ( n ) ( x ) = ( n x ) p x ( 1 p ) n x where x = , 1 , 2 ,..., n ( 1 p + zp ) n Geometric p X ( x ) = p...
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 Fall '07
 Carlton

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