Example of Dependent but Uncorrelated Random VariablesDate: 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/ |
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