Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.24 GENERATING FUNCTIONS

In this section, we consider expectations similar to the CF that can also be used to generate moments as well as cumulants.

Definition: Moment Generating Function The moment generating function (MGF) of random variable X is

(5.318)

for and with . The nth moment is generated as follows:

(5.319)

where m(n)X(t) is the nth ordinary derivative of mX(t) with respect to t.

The MGF does not always exist whereas the CF usually does. Recall that the CF includes in the exponent of the exponential , and is similar to the Fourier transform. The moment-generating property of mX(t) is easily verified by expanding the last expression in (5.318):

(5.320)

For example, differentiating twice gives

(5.321)

Obviously, terms subsequent to can be nonzero for because of the higher ...