In a roll of die, if $X$ is the number on the first die and $Y$ is the number on second die, then determine whether the random variable $X+Y$ and $X-Y$ are independent.
The covariance between the two turned out to be $\mathrm{Var}(X) + \mathrm{Var} (Y)$. So zero covariance would mean that $\mathrm{Var}(X)$ and $\mathrm{Var}(Y)$ is zero, that is, no spread. But we also know that zero covariance does not imply independence. I really cannot think of a way to prove independence between the two.
Best Answer
They're not: If $X+Y=12$ then both rolls were sixes, so $X-Y=0$. So you have:
$$1 = \mathbb{P}(X-Y =0|X+Y=12) \neq \mathbb{P}(X-Y =0) = \frac{1}{6}.$$