Are the statements about the correlation coefficient true or false?

Question

Let us define the correlation coefficient as $\rho(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$.

Are the following statements true or false?

1. If $\rho(X,Y)=\rho(Y,Z)=0$ then $\rho(X,Z)=0$

2. If $\rho(X,Y)>\rho(Y,Z)>0$ then $\rho(X,Z)>0$

3. If $\rho(X,Y)<\rho(Y,Z)<0$ then $\rho(X,Z)<0$

I think they are false, but I can't find counterexamples. Could you help me?

Answer 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 $N(0,1)$ random variables and $Y:= X + Z$. Basically the same counterexample works for the third, but with $Y := -X-Z$ instead.