I encountered 2 questions regarding independence of 2 random variables.
1) A and B are i.i.d normal and independent, Let $X = 3A + 2B$ and $Y = 2A – 3B$, prove that they are independent
2) Will they still be independent if the normality condition is dropped?
1) I assume I should notice that linear combination of i.i.d normal variables forms a jointly normally distributed variables X and Y, which means, that if the covariance is 0, then the independence is proven. Covariance ends up being 0.
2) I'm really unsure about how to approach this one. I have a hunch that the answer is negative but I'm simply bad at coming up with some simple examples.
I'd appreciate and insight into the approaches for such questions.
Best Answer
Your answer to question 1 is correct. For question 2, try the following $A$ and $B$ are i.i.d., take the values $0$ and $1$ with probability $1/2$. Then the vector $(X,Y)$ takes the values $(0,0)$, $(3,2)$, $(2,-3)$ and $(5,-1)$ with probability $1/4$. Let $E_1:= \{X=0\}$ and $E_2:= \{Y=2\}$. These events are not independent because their intersection have probability $0$ while both of them have probability $1/4$.