TITLE: Motivation of Moment Generating Functions
QUESTION [8 upvotes]: What is the motivation of defining the moment generating function of a random variable $X$ as: $E[e^{tX}]$? I know that one can obtain the mean, second moment, etc.. after computing it. But what was the intuition in using $e^{x}$? Is it because its one-to-one and always increasing?

REPLY [6 votes]: The goal is to to put all the moments in one package. Since
$$
 e^{tx} = \sum \frac{x^n}{n!} t^n
$$
the coefficients of $t^n$ in $E(e^{tx})$ are (scaled) moments. In other contexts we can use
$$
(1-xt)^{-1} = \sum x^n t^n
$$
in place of $e^{tx}$. This gives more or less what engineers call the "z-transform"
and in combinatorics it is known as "ordinary generating function". Using the exponential
has the happy advantage that convolution of random variables translates to product
of moment generating functions.