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Suppose we have random variables~$X,Y,Z$ that are related by the following:

\begin{equation} X = Y + f(Z) \end{equation}

for some function $f$. Under what conditions on the random variables and $f$ can we always say that $X$ and $Y$ are positively correlated? Or is it always the case given the linear relation?

Thanks.

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We always have the following relation: $$cov(X,Y)=var(Y)+cov(Y,f(Z))$$

Assuming $f$ is a deterministic function, if $Y$ and $Z$ are independent, the correlation is nonnegative since $cov(X,Y)=var(Y)\geq 0$. It is strictly positive if $Y$ is not a constant random variable.

If $Y$ and $Z$ are somehow dependent, we need to have the following inequality for positive correlation: $$cov(Y, f(Z))> -var(Y)$$

We can list some simple cases like $f$ is constant function, or $f(Z)=Y$, but I don't think there is more to say about it.

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