Answer

Solution

1. 1

   Zero correlation is by definition the absence of any linear association.

2. 2

   Counterexamples eliminate the other options:

3. 3

   \begin{itemize}

4. 4
   • Zero correlation does not imply independence (e.g.\ $Y = X^2$ with $X$ symmetric about $0$ yields zero correlation but $X, Y$ are dependent).
5. 5
   • In the same example, $X$ and $Y$ are functionally related.
6. 6

   \end{itemize}

7. 7

   Thus only ``no linear relationship'' is always true.

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