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# More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

## 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 discrete-time Fourier transform, and as before is the ''frequency'' variable. (Comment on the sign of the exponent.) Given a , we can find by the inverse discrete-time Fourier transform.

Properties:

1. . (Why?)
2. . (Why?)
3. and form a unique Fourier transform pair.

Thus, provides yet another way of displaying the probability structure of .
4. . This is referred to as the Fourier-Stieltjes transform of .
5. is uniformly continuous.

We can write

Theorem 1   If then

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 (two-dimensional Fourier transforms).

Properties:

1. Moments:

2. 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.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec2_6.html. This work is licensed under a Creative Commons License