# Null covariance between X and Y: non-linear relationship between them

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 $$P(x,y)$$: 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?

## 1 Answer

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.

• Thank you for your answer! I edited my post... – Mark Apr 9 at 10:37
• Can we prove that the relationship is non linear? I mean, even without knowing this relation... – Mark Apr 10 at 18:42
• What do you mean by non-linear: $Y\neq aX+b$ ? – Geethu Joseph Apr 11 at 3:53
• Yes I mean this. – Mark Apr 11 at 10:20
• 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. – Geethu Joseph Apr 11 at 10:32