## Moment-Generating Functions and Characteristic Functions

2.8.1. Moment-Generating Functions

The moment-generating function ofa bounded random variable X (i. e., P [| X | < M] = 1 for some positive real number M < to) is defined as the function

m(t) = E[exp(t ■ X)], t e R, (2.31)

where the argument t is nonrandom. More generally:

Definition 2.15: The moment generating function of a random vector X in R* is defined by m(t) = E[exp(tTX)] for t e T c R*, where T is the set of nonrandom vectors t for which the moment-generating function exists and is finite.

For bounded random variables the moment-generating function exists and is finite for all values of t. In particular, in the univariate bounded case we can write

It is easy to verify that the jth derivative of m(t) is

dj m(t) ^ t*—j E[X*]

(dt )j = *=j (* — j)!

hence, th...

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