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

In summary: Covariance is zero only if one of the variables has zero mean.In summary, the conversation discusses the correlation between XY and Y in terms of the means and standard deviations of X and Y. It is mentioned that if X and Y are independent, then X and Y^2 are also independent. However, the formula for covariance shows that it is not always zero unless E(X)=0.
  • #1
rhuelu
17
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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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  • #2


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.
 

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

What is correlation and how is it calculated?

Correlation is a statistical measure that shows the degree of relationship between two variables. It is calculated using a formula called the correlation coefficient, which ranges from -1 to 1. A positive correlation means that the two variables move in the same direction, while a negative correlation means they move in opposite directions.

Why is it important to study the correlation between XY and Y?

Studying the correlation between XY and Y can help us understand the relationship between these two variables and how they affect each other. This information can be useful in making predictions, identifying patterns, and developing theories in various fields of study, such as economics, psychology, and biology.

Is correlation the same as causation?

No, correlation does not imply causation. Just because two variables are correlated, it does not mean that one causes the other. There could be other factors at play that influence both variables, or the relationship could be coincidental.

How do outliers affect correlation?

Outliers, which are extreme values in a dataset, can greatly affect the correlation between two variables. They can either strengthen or weaken the correlation, depending on where they fall in relation to the other data points. It is important to identify and deal with outliers appropriately when studying correlation.

Can correlation be used to make predictions?

Yes, correlation can be used to make predictions, but it should be done with caution. While a strong correlation may suggest a relationship between two variables, it does not necessarily mean that one causes the other. Other factors should be taken into consideration when making predictions based on correlation.

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