Is the Correlation between XY and Y Zero if X and Y are Independent?

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SUMMARY

The correlation between the product XY and Y is not necessarily zero, even if X and Y are independent. The covariance formula cov(XY, Y) = E(XY^2) - E(X)E(Y)^2 indicates that the correlation depends on the expected values of X and Y. Specifically, if E(X) is not zero, then cov(XY, Y) will not equal zero, leading to a non-zero correlation. Therefore, while X and Y are independent, the independence does not extend to XY and Y.

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rhuelu
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I would appreciate some help with this problem. Assuming X and Y are independent, I'm trying to find the correlation between XY and Y in terms of the means and standard deviations of X and Y. I'm not sure how to simplify cov(XY,Y)=E(XYY)-E(XY)E(Y)
=E(XY^2)-E(X)E(Y)^2.

If X and Y are independent, does it follow that X and Y^2 are independent. If this is the case, then covariance is zero --> correlation is zero. If this isn't the case I'm really not sure how to proceed. Any help is appreciated...
 
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rhuelu said:
I would appreciate some help with this problem. Assuming X and Y are independent, I'm trying to find the correlation between XY and Y in terms of the means and standard deviations of X and Y. I'm not sure how to simplify cov(XY,Y)=E(XYY)-E(XY)E(Y)
=E(XY^2)-E(X)E(Y)^2.

If X and Y are independent, does it follow that X and Y^2 are independent. If this is the case, then covariance is zero --> correlation is zero. If this isn't the case I'm really not sure how to proceed. Any help is appreciated...
X and Y^2 are independent. However your formula has cov(XY,Y)=E(X)[E(Y^2)-E(Y)^2] which is not 0, unless E(X)=0.
 

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