2022-07-20

Correlation of Rolling Two Dice
If A is a random variable responsible for calculating the sum of two independent rolls of a die, and B is the result of calculating the value of first roll minus the value second roll, is is true that A and B have a $cov\left(A,B\right)\ne 0$? In other words, is it true that they are correlated?
I've come to the conclusion that they must be correlated because they are not independent, that is, the event of A can have an impact on event B, but I remain stuck due to the fact that causation does not necessarily imply correlation.
I know that independence −> uncorrelation, but that the opposite isn't true.

iljovskint

Covariance of independent variables is 0 but covariance of dependent variables is not necessarily non-zero: it might be 0 (which is exactly what happens in this case), so your conclusion is untrue.
Let ${X}_{1},{X}_{2}$ be independent random variables denoting the number rolled on the two fair die respectively. ${X}_{1},{X}_{2}$ are identically distributed. $A={X}_{1}+{X}_{2},B={X}_{1}-{X}_{2}$
$\mathbb{E}\left[B\right]=E\left[{X}_{1}\right]-E\left[{X}_{2}\right]=0.\phantom{\rule{0ex}{0ex}}\mathbb{E}\left[AB\right]=E\left[{X}_{1}^{2}\right]-E\left[{X}_{2}^{2}\right]=0.$
So the covariance $\mathbb{E}\left[AB\right]-\mathbb{E}\left[A\right]\mathbb{E}\left[B\right]$ is 0.

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