Zero Correlation Does Not Imply Independence

Date: 11/04/2003 at 09:30:36
From: Henna
Subject: zero correlation does not imply independence

I understand that if 2 random variables are independent, then their 
correlation is zero and I have seen many examples of this.  However, 
I can't find any examples of when two random variables have zero 
correlation, yet are not independent. 

Date: 11/04/2003 at 18:00:56
From: Doctor Douglas
Subject: Re: zero correlation does not imply independence

Hi Henna.

Thanks for writing to the Math Forum.

Here is a simple example where the two random variables have zero
correlation, yet are not independent. 

  Suppose X is a normally-distributed random variable with 
  zero mean.  Let Y = X^2.  Clearly X and Y are not independent:  
  if you know X, you also know Y.  And if you know Y, you know the
  absolute value of X.

  The covariance of X and Y is

     Cov(X,Y) = E(XY) - E(X)E(Y) = E(X^3) - 0*E(Y) = E(X^3)
     
              = 0, 
  
  because the distribution of X is symmetric around zero.  Thus
  the correlation r(X,Y) = Cov(X,Y)/Sqrt[Var(X)Var(Y)] = 0, and
  we have a situation where the variables are not independent, yet
  have (linear) correlation r(X,Y) = 0.

This example shows how a linear correlation coefficient does not
encapsulate anything about the quadratic dependence of Y upon X.  

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

Date: 11/04/2003 at 18:16:22
From: Henna
Subject: Thank you 

Thank you very much, Dr Douglas.  The example that you have given me
has certainly made things seem clear!