## 2.10.4 The s-TransformFor continuous random variables the s-transform plays a role analogous to that played by the z-transform with discrete random variables. For a random variable X with pdf f _{x}(x), the s-transform (or exponential
transform or Laplace transform) is defined to beIf s is considered to be a complex number, with real and imaginary parts, the integrals associated with s-transforms and their derivatives will be finite if the real part of s is zero. If P{X a} = 0 for some finite a, these integrals will be finite as long as the real part of s is nonnegative. In the remainder of our work we will assume that the real part of s is chosen appropriately. And again, as with z-transforms, emphasis in subsequent chapters will not be given to transform techniques. Given the s-transform of a pdf, one can uniquely recover the pdf. In general, this is done by contour integration in the complex plane. For those s-transforms whose numerators and demoninators factor into products of terms (s - s _{1})(s - s_{2}) . . ., the pdf can be recovered by partial
fraction expansion.By direct substitution into the definition of the s-transform, one can verify the following moment-generating properties: Applications of these relationships are shown in the following section. Exercise 2.14: s-Transform of a Sum Suppose that X_{1},
X_{2}, . . ., X_{n}.
are mutually independent continuous random variables.
Let R = X_{1} + X_{2} + . . . + X_{n}. Show that |