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Let us consider a variable X with values $\{-1,1\}$, and another one Y with values $\{-2,-1,1,2\}$.

X and Y have the following joint probability function $P(x,y)$:

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From this table, the covariance is zero.

Does this mean that there are no linear relationships between X and Y (because covariance function should measures linear relationships)?

Then, is it possible to calculate the non linear relation between X and Y or, at least, prove that their relationship is non linear?

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In this case, $y$ takes the values $\pm\left(\frac{3}{2}x-\frac{1}{2}\right)$ with equal probabilities. Since $y$ take both positive and negative values with equal probabilities, we can't say that $y$ increases or decreases with $x$. However, it is not possible to deduce such relationships easily, for a general case.

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  • $\begingroup$ Thank you for your answer! I edited my post... $\endgroup$ – Mark Apr 9 at 10:37
  • $\begingroup$ Can we prove that the relationship is non linear? I mean, even without knowing this relation... $\endgroup$ – Mark Apr 10 at 18:42
  • $\begingroup$ What do you mean by non-linear: $Y\neq aX+b$ ? $\endgroup$ – Geethu Joseph Apr 11 at 3:53
  • $\begingroup$ Yes I mean this. $\endgroup$ – Mark Apr 11 at 10:20
  • $\begingroup$ Such a relation holds only if for every value of $X$, there exists a unique value of $Y$. For a general case where there is a one-to-one correspondence between $X$ and$Y$, it is easy to verify if a such linear relation holds. $\endgroup$ – Geethu Joseph Apr 11 at 10:32

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