Let $X$ and $Y$ be random variables that represent the result of a thrown of two dice ($X$ represents the result of the first dice, $Y$ of the second one. In other words, $\mathbb{P}[X=k]=\mathbb{P}[Y=k]=1/6$ for every $k=1,\dots,6$). Tell if $X-2Y$ and $X+Y$ are independent or not.
This is what I've done: $$\mathbb{P}[X-2Y=k]\mathbb{P}[X+Y=h]=\left(\sum_{j=1}^6p_X(k+2j)p_Y(j)\right)\left(\sum_{j=1}^6p_X(h-j)p_Y(j)\right)$$ While: $$\mathbb{P}[X-2Y=k,\ X+Y=h]=\sum_{j=1}^6 \mathbb{P}[X=k+2j,\ X=h-j,\ Y=j]=\sum_{j=1}^6\mathbb{P}[X=k+2j,\ X=h-j]\mathbb{P}[Y=j]$$ Then I don't know how to proceed, and honestly I'm not even sure that what I've done is totally correct.
They're not: $P(X + Y = 12) = P(X = 6, Y = 6) = 1/36$. But if $X = 6$ and $Y = 6$ then $X - 2Y = -6$, so
$$P(\{X + Y = 12\} \cap \{X - 2Y = 0\}) = 0 \neq P(X + Y = 12) \cdot P(X - 2Y = 0)$$