Question: You are given a fair red die and a fair blue die. You roll each die once, independently of each other. Let $(i, j)$ be the outcome, where $i$ is the result of the red die and $j$ is the result of the blue die. Define:
$X = i+j$ and $Y = i-j$
Are these independent or not?
Attempt:
Just by looking at it, it seems that they are dependent on one another but I’m not sure how to prove it.
If I take, $i=6$ and $j=4$ then $X=10$, and $Y =2$.
I think I need to use this formula: $Pr$ ($X=x$ $\land$ $Y=y$) $=$ $Pr(X=x) $ . $ Pr(Y=y)$
Wouldn’t the probability be $\frac{1}{6}$ for each one? Not sure how to use the formula to prove dependence.
If you want to show that two random variables are not independent, you would show that there are $x,y$ such that $\Pr(X = x \land Y = y) \neq \Pr(X = x) \cdot \Pr(Y = y)$. Here, there's a lot of choices you can take. For example, an easy one would be $x = 12, y = -5$. Note that both $X = 12$ and $Y = -5$ can happen with probability $\frac{1}{36}$: the first requires rolling two sixes, and the second requires rolling a 1 then a 6. But it is impossible for both two happen together. Hence, $$\Pr(X = x \land Y = y) = 0 \neq \frac{1}{36} \times \frac{1}{36} = \Pr(X = x) \cdot \Pr(Y = y)$$ So the two are not independent. Of course, there are many choices of $x$ and $y$ where this relationship fails to holds; any such pair suffices.