# Are the statements about the correlation coefficient true or false?

Let us define the correlation coefficient as $\rho \left(X,Y\right)=\frac{Cov\left(X,Y\right)}{\sqrt{Var\left(X\right)Var\left(Y\right)}}$ .
Are the following statements true or false?
If $\rho \left(X,Y\right)=\rho \left(Y,Z\right)=0$ then $\rho \left(X,Z\right)=0$
If $\rho \left(X,Y\right)>\rho \left(Y,Z\right)>0$ then $\rho \left(X,Z\right)>0$
If $\rho \left(X,Y\right)<\rho \left(Y,Z\right)<0$ then $\rho \left(X,Z\right)<0$
I think they are false, but I can't find counterexamples. Could you help me?
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Marshall Horne
Step 1
They are indeed all false. For the first one, you can take $X=Z$ as a counterexample, and have Y be independent of X. For the second, you can take X and Z to be iid $N\left(0,1\right)$ random variables and $Y:=X+Z$ . Basically the same counterexample works for the third, but with $Y:=-X-Z$ instead.