Let both $X$ and $Y$ be independent and standard normal distributed random variables with mean $0$ and variance $1$. What is the distribution of:
I know that both $X+Y$ and $X-Y$ have the same distribution; they're both normal with mean $0$ and variance $2$. But I'm not sure how to describe the ratio of the two; furthermore, these normal random variables that we're taking the ratio of aren't independent either, since they're both composed of $X$ and $Y$. Finally, I'm not sure how the absolute value changes things (it makes the denominator a "folded normal" distribution, but I'm not sure how to work with that in this context).
Similarly, what is the distribution of:
I end up with the ratio of two separate, non-independent chi square random variables with 1 degree of freedom. How do I describe the distribution of that?
So what are the distributions of $Q$ and $R$?