Example of Dependent but Uncorrelated Random Variables

Date: 01/25/2006 at 01:56:06
From: George
Subject: Example of dependent but uncorrelated random variables

One reads often that two independent random variables are always
uncorrelated but that the converse is not always true.  Can you
provide an example of two random variables that are uncorrelated but
NOT independent?

Date: 01/25/2006 at 10:39:46
From: Doctor George
Subject: Re: Example of dependent but uncorrelated random variables

Hi George,

Thanks for writing to Doctor Math.

Let X be normally distributed with E(X) = 0.  Also, let Y = X^2.  Thus
X and Y are plainly not independent since

   f(y|x) <> f(y)
    Y         Y

However,

   Cov(XY) = E(XY) - E(X)E(Y)

           = E(X^3) - 0

           = 0

Therefore X and Y are uncorrelated, but not independent.  The key is
that correlation is only a measure of linear dependence.  It does not
necessarily imply anything about other kinds of dependence.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/26/2006 at 00:58:31
From: George
Subject: Thank you (Example of dependent but uncorrelated random
variables)

Thank you.  Excellent example, because it is so simple!  I would 
simplify it a bit more by saying that Y = X^2 and X is a random 
variable with vanishing mean, finite 2nd moment, and vanishing 3rd
moment.  X does not have to be normally distributed.  Any density
function that is symmetric about 0 and for which Integral(|x|^3 dP)
exists will do.

Date: 01/26/2006 at 06:58:55
From: Doctor George
Subject: Re: Thank you (Example of dependent but uncorrelated random
variables)

Hi George,

I'm glad that you were able to follow my reasoning.

You are correct that X does not have to be normally distributed.  I
simply made that choice because it is familiar and has the necessary
properties that you listed.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/