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Assume there are $3$ random variables $X$, $Y$ and $Z$ such that $\operatorname{corr}(X,Y) = 0.5$, $\operatorname{corr}(X,Z) = -0.5$. What is the exact range for $\operatorname{corr}(Y, Z)$?

My approach: I tried using the formula for correlation i.e.

$$\frac{E[XY]-E[X]E[Y]}{\operatorname{std}(X)\operatorname{std}(Y)}$$

and substituting the variables but I didn't reach anywhere. Can someone suggest a way? I was looking at various explanations of similar type problems on the net and found they have certain relation with semidefinite matrix. I have no idea what they are. If they are useful, I would encourage you to kindly leave a link too so that I can learn it. Thanks!

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From your assumptions it follows that there exist, two random variables $X^\bot,X^{\bot\bot}$ that are both uncorrelated with $X$ and have variance one such that \begin{align} Y&=c_{12}\,X+\sqrt{1-c_{12}^2}\,X^\bot\\ Z&=c_{13}\,X+\sqrt{1-c_{13}^2}\,X^{\bot\bot}\\ \end{align} where $c_{12}=1/2$ and $c_{13}=-1/2\,$. Writing $$ \operatorname{corr}(X^\bot,X^{\bot\bot})=\rho $$ we have

$$ \operatorname{corr}(Y,Z)=c_{12}\,c_{13}+\sqrt{1-c_{12}^2}\,\sqrt{1-c_{13}^2}\,\rho\,. $$ The range of this correlation is obtained from $\rho\in[-1,1]$.

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  • $\begingroup$ I don't get it, we had some options it was a MCQ, a few of them being [-1,0), [-1,0.5] and so on. How can we even get such an exact range from this formula you have written $\endgroup$
    – Charlie
    Aug 11, 2022 at 4:03
  • $\begingroup$ How about "plugging in" your values for $c_{12},c_{13}$ as they are $\pm 1/2$ ? $\endgroup$
    – Kurt G.
    Aug 11, 2022 at 6:15
  • $\begingroup$ ah, sorry. I only saw the final formula accepted your answer now :) $\endgroup$
    – Charlie
    Aug 11, 2022 at 6:17
  • $\begingroup$ Great. Can you figure out if this gives the same answer as Sylvestor's criterion that was kindly mentioned by @mark leeds in his comment above ? $\endgroup$
    – Kurt G.
    Aug 11, 2022 at 6:19

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