Let us define the correlation coefficient as ρ ( X , Y ) = C o v ( X , Y ) V a r ( X ) V a r ( Y ) .Are the following statements true or false?If ρ ( X , Y ) = ρ ( Y , Z ) = 0 then ρ ( X , Z ) = 0If ρ ( X , Y ) &gt; ρ ( Y , Z ) &gt; 0 then ρ ( X , Z ) &gt; 0If ρ ( X , Y ) &lt; ρ ( Y , Z ) &lt; 0 then ρ ( X , Z ) &lt; 0I think they are false, but I can't find counterexamples. Could you help me?

Question

Let us define the correlation coefficient as   ρ  (  X  ,  Y  )  =            C      o      v      (      X      ,      Y      )              V      a      r      (      X      )      V      a      r      (      Y      )       .Are the following statements true or false?If   ρ  (  X  ,  Y  )  =  ρ  (  Y  ,  Z  )  =  0 then   ρ  (  X  ,  Z  )  =  0If   ρ  (  X  ,  Y  )  &amp;gt;  ρ  (  Y  ,  Z  )  &amp;gt;  0 then   ρ  (  X  ,  Z  )  &amp;gt;  0If   ρ  (  X  ,  Y  )  &amp;lt;  ρ  (  Y  ,  Z  )  &amp;lt;  0 then   ρ  (  X  ,  Z  )  &amp;lt;  0I think they are false, but I can&#039;t find counterexamples. Could you help me?

Marshall Horne · Accepted Answer

Step 1They 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.

Are the statements about the correlation coefficient true or false?

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