2.8 THE z-TRANSFORMThe expectation of one particular function g(X), namely g(X) = zx, is Of wide use in computations and analysis for those discrete random variables X that take on only nonnegative integer experimental values. This expected value is defined to be the z-transform (or discrete transform or geometric transform or moment-generating function) of px(x), ![]() Since 0 ![]() ![]() Given the z-transform of a pmf, we can uniquely recover the pmf. We do this by considering the definition of the z-transform, ![]() The important moment-generating properties of the z-transform are obtained from the following relationships: ![]() Applications of these relationships are shown in the following section. Exercise 2.10: z-Transform of a Sum Suppose that X1, X2,. . . , Xn. are mutually independent random variables. Let S = X1 + X2 + ...+ Xn . Show that ![]() |