From Algorithms to Z-Scores - matloff - University of California, Davis

19.4. TRANSFORM METHODS 387
19.4.2 Moment Generating Functions
The generating function is handy, but it is limited to discrete random variables. More generally,
we can use the moment generating function, defined for any random variable X as
for any t for which the expected value exists.
mX(t) = E[e tX ] (19.56)
That last restriction is anathema to mathematicians, so they use the characteristic function,
φX(t) = E[e itX ] (19.57)
which exists for any t. However, it makes use of pesky complex numbers, so we’ll stay clear of it
here.
Differentiating (19.56) with respect to t, we have
We see then that
m ′ X(t) = E[Xe tX ] (19.58)
m ′ X(0) = EX (19.59)
So, if we just know the moment-generating function of X, we can obtain EX from it. Also,
so
m ′′ X(t) = E(X 2 e tX ) (19.60)
m ′′ X(0) = E(X 2 ) (19.61)
In this manner, we can for various k obtain E(X k ), the kth moment of X, hence the name.