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### 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

- 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!
Associated Topics:
College Statistics
High School Statistics

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