Are the statements about the correlation coefficient true or false?

garnirativ8 2022-09-30 Answered
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 ) = 0
If ρ ( X , Y ) > ρ ( Y , Z ) > 0 then ρ ( X , Z ) > 0
If ρ ( X , Y ) < ρ ( Y , Z ) < 0 then ρ ( X , Z ) < 0
I think they are false, but I can't find counterexamples. Could you help me?
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Answers (1)

Marshall Horne
Answered 2022-10-01 Author has 8 answers
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 ( 0 , 1 ) random variables and Y := X + Z . Basically the same counterexample works for the third, but with Y := X Z instead.
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