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I came across following problem:

A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$?

I solved it as follows:

 P = -3, -2, 0, 2, 3    Mean = 0/0 = 0
 Q =  9,  4, 0, 4, 9    Mean = 26/5 = 5.2

 Covariance = ((-3)(3.8) + (-2)(-1.2) + (0)(-5.2) + (2)(-1.2) + (3)(3.8)) / 5
            = 0 / 5 
            0

So I was guessing what does the covariance of zero here means. Does that mean that P and Q do not co-variate (deviate together) at all? Or my approach or calculations were incorrect?

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Your result is correct. While independent variables will have a covariance of 0, 0 covariance does not guarantee independence - it measures the linear relationship, and in this case the relationship is non-linear.

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  • $\begingroup$ how to detect and measure the non-linear relationship that covariance can't? $\endgroup$ – develarist Aug 23 '20 at 20:33
  • $\begingroup$ There are a few suggestions here. In this case I'd probably try distance correlation, from the third answer. stats.stackexchange.com/questions/35893/… $\endgroup$ – Elenchus Aug 23 '20 at 21:41

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