Covariance between a variable and an interaction variable

by Gilean0709   Last Updated July 28, 2015 13:08 PM

If I have two independent variables $X$ and $Y$, then $Cov(X,Y)=0$. Now let $Z = X*Y$. Then I would assume $Cov(X,Z)\ne 0$, but given the expecations, variances and covariances of $X$ and $Y$ is there are formula I can use to calculate how big it should be?

Answers 1

I am not sure if I am clear about your question. \begin{align} Cov(X,Z)&=Cov(X,XY)\\ &=E(X^2Y)-E(X)E(XY)\\ \text{(because of independence)} &= E(X^2)E(Y)-E^2(X)E(Y)\\ &=E(Y)Var(X) \end{align}

July 28, 2015 08:59 AM

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