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Characteristic functions
The characteristic function is essentially the Fourier transform of the p.d.f. or p.m.f. They are useful in practice not for the usual reasons engineers use Fourier transforms (e.g., frequency content), but because they can provide a means of computing moments (as we will see), and they are useful in finding distributions of sums of independent random variables.
Let us write some more explicit formulas. Suppose
is a continuous
random variable. Then (by the law of the unconcious statistician)
This may be recognized as the Fourier transform of , where is the ''frequency'' variable. (Comment on sign of exponent.) Note that given we can determine by an inverse Fourier transform:
If
is a discrete r.v.,
which we recognize as the discretetime Fourier transform, and as before is the ''frequency'' variable. (Comment on the sign of the exponent.) Given a , we can find by the inverse discretetime Fourier transform.
Properties:
 . (Why?)
 . (Why?)

and
form a unique Fourier transform pair.
 . This is referred to as the FourierStieltjes transform of .
 is uniformly continuous.
We can write
That is, we can obtain moments by differentiating the characteristic function. For this reason, characteristic functions (or functions which are very similarly defined) are sometimes referred to as moment generating functions .
Then
and
are uniquely related (twodimensional
Fourier transforms).
Properties:

Moments:
 and are independent if and only if for all .
Sums of independent random variables
Let
and
be independent r.v.s, and let
Then
But also
So
If and are continuous r.v.s, then so is .
by the convolution theorem .
Thus, when continuous independent random variables are added, the p.d.f of the sum is the convolution of the p.d.f.s (and respectively p.m.f. for discrete independent r.v.s).
An example: Jointly Gaussian
If
, then
We make an observation here: the ''form'' of the Gaussian p.d.f. is the exponential of quadratics. The form of the Fourier transform of the eponential of quadratics is of the form exponential of quadratics. This little fact gives rise to much of the analytical and practical usefulness of Gaussian r.v.s.
Characteristic functions marginals
We observe that
In our Gaussian example, we have
which is the ch.f. for a Gaussian,
We could, of course, have obtained a similar result via integration, but this is much easier.