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

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If X and Y are independent, the correlation between XY and Y can be analyzed through covariance, specifically cov(XY, Y) = E(XY^2) - E(X)E(Y)^2. The discussion raises the question of whether X and Y^2 are also independent, which is affirmed. However, it is noted that the covariance formula does not yield zero unless E(X) equals zero, indicating that correlation is not necessarily zero. Thus, while independence suggests a lack of correlation, the specific conditions of the means and variances must be considered for accurate conclusions.
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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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