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In a Bioinformatics article appendix, I found the equation :

$$ Cov(X/Z, Y/Z) = Cov(X, Y)E(1/Z^2) + E(X)E(Y)Var(1/Z) $$

when supposing that the random vector $(X, Y)$ with covariance $Cov(X, Y)$ is independent of the random variable $Z$. $X$ and $Y$ are not correlated. What are the steps to arrive at that expansion ?

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$Cov(X/Z, Y/Z)=E(\frac{XY}{Z^2})-E(\frac{X}{Z})E(\frac{Y}{Z}).$ The independence of the random vector $(X,Y)$ of $Z$ implies that the vector is also independent of $\frac{1}{Z}$. Then $=E(\frac{XY}{Z^2})-E(\frac{X}{Z})E(\frac{Y}{Z})=E(XY)E(\frac{1}{Z^2})-E(X)E(Y)E(\frac{1}{Z})^2$.

We can add and substract the same element without changing the expression:

$=E(XY)E(\frac{1}{Z^2})- [E(X)E(Y)E(\frac{1}{Z^2})-E(X)E(Y)E(\frac{1}{Z^2})] -E(X)E(Y)E(\frac{1}{Z})^2=$ $=E(\frac{1}{Z^2})[E(XY)-E(X)E(Y)] + E(X)E(Y)[E(\frac{1}{Z^2}) - E(\frac{1}{Z})^2]=$ $=E(\frac{1}{Z^2})Cov(X,Y) + E(X)E(Y)Var(\frac{1}{Z})$.

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