The characteristic function is the expectation of a particular complex-valued function of a random variable. It is useful for systematically determining moments of a random variable, as well as for computing functions of variables such as the sum of two independent random variables covered in Chapter 4.

Definition: Characteristic Function The characteristic function (CF) of random variable X is

(5.85) Numbered Display Equation

where and .

Substituting into (5.80) gives

(5.86) Numbered Display Equation

which we see is similar to the Fourier transform of the pdf (the exponent is positive instead of negative as in the usual definition of the Fourier transform). Like the Fourier transform, the following conditions are sufficient for to exist on the support of FX(x):

  • Finite number of discontinuities.
  • Bounded variation (see Appendix B).
  • Absolutely integrable:


Note that all the distributions summarized ...

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