Zero Correlation Does Not Imply IndependenceDate: 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! |
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