# Is it possible to determine the relationship between X and Y, with only X and Z, where Z = X + Y?

I have a set of data composed of two variables $X$ and $Z$, where $Z = X + Y$. I want to make a statement about the relationship between $X$ and $Y$. For instance, I'd like to claim that $X$ and $Y$ vary together or correlate. Is there a way to do this with only measurements from $X$ and $Z$?

As an additional wrinkle, consider if X and Z are corrupted by some noise $X$-noise and $Z$-noise. $X$ is actually $X + X$-noise and $Z$ is $X + Y + Z$-noise. In addition, $X$ is scaled by some coefficient $a_X$ and $Z$ is scaled by some other coefficient $a_Z$. Therefore, the data I collected, which are paired measurements, are $[a_X*(X + X-\text{noise}), a_Z*(X + Y + Z-\text{noise})]$. Can I still estimate $Y$ from my measurements? If I do not know $a_X$ or $a_Z$ is this problem unsolvable?

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$Y=Z-X$ which should give you something to analyse – Henry Sep 28 '12 at 19:59
That seems to be the answer, @Henry, care to make it official? – gung Sep 28 '12 at 20:05
@gung: I have a slight suspicion that I am not answering the question wherestheforce intended to ask. – Henry Sep 28 '12 at 20:07
@Henry it looks like it deals with the question to me – Glen_b Sep 29 '12 at 1:37
Cov(X,Y)=Cov(X,Z-X)=Cov(X,Z)-Var(X). Fron data on pairs of values X and Z you can estimate Cov(X,Z) and Var(X). Is that what Henry was getting at? Is it fair to assume when the OP says "we have measurements for X and Z" that the data is paired? If not I don't see how covariances or correlations can be estimated. – Michael Chernick Sep 29 '12 at 2:29
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