Correlation pattern

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!

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 $$\rho\in[-1,1]$$.
• How about "plugging in" your values for $c_{12},c_{13}$ as they are $\pm 1/2$ ? Aug 11, 2022 at 6:15