58.17 Use Of Characteristic Functions To Find Moments
Let X be a random variable with characteristic function
Then this can be used to find moments of the random variable assuming they exist. The
kth moment is defined as
This can be done by using the dominated convergence theorem to differentiate
the characteristic function with respect to t and then plugging in t = 0. For
and now plugging in t = 0 you get iE
Doing another differentiation you
and plugging in t = 0 you get −E
and so forth.
An important case is where X is normally distributed with mean 0 and variance σ2.
In this case, as shown above, the characteristic function is
Also all moments exist when X is normally distributed. So what are these moments?
and plugging in t = 0 you find the mean equals 0 as expected.
and plugging in t = 0 you find the second moment is σ2. Then do it again.
and so E
By now you can see the pattern. If you continue this way, you find
the odd moments are all 0 and
This is an important observation.