In a paper I've written I model the random variables $X+Y$ and $X-Y$ rather than $X$ and $Y$ to effectively remove the problems that arise when $X$ and $Y$ are highly correlated and have equal variance (as they are in my application). The referees want me to give a reference. I could easily prove...
I was hoping someone could propose an argument explaining why the random variables $Y_1=X_2-X_1$ and $Y_2=X_1+X_2$, $X_i$ having the standard normal distribution, are statistically independent. The proof for that fact follows easily from the MGF technique, yet I find it extremely counter-intui...
The latter question has received a few more detailed answers, but IMO Dmitrij Celov's comment to Rob's question is what immediately springs to my mind. Whether it is intuitive or not I don't know.
@AndyW, so you're wondering if you should vote to close the older Q? I would put a comment w/ a link on both, but not vote to close. Rob's Q asks for a reference & the new Q asks for the intuition, so they may be just far enough apart.
I was hoping someone could propose an argument explaining why the random variables $Y_1=X_2-X_1$ and $Y_2=X_1+X_2$, $X_i$ having the standard normal distribution, are statistically independent. The proof for that fact follows easily from the MGF technique, yet I find it extremely counter-intui...
