# Finding distribution and checking independence of transformed normal variables

$$X,Y,Z$$ are three independent random variables following standard normal distribution. Consider a real function $$f$$ such that \begin{align}f(x)&=1 , x\geq 0 \\ &= -1, x<0\\ \end{align} Let $$U,V,W$$ be defined such that \begin{align}U&=|X| f(Y)\\ V&=|Y| f(X)\\ W&=|Z| f(X)\\ \end{align} Then, how do I check the pairwise and mutual independence of $$U,V,W$$ and that these follow normal distribution.

I have no clue on how to proceed. The modulus of random variables makes it even more confusing.

• What ways do you know of characterizing independence? Which have you tried to apply? – whuber May 28 '19 at 16:13
• It is $U=|X| f(Y), V=|Y| f(X), W=|Z| f(X)$. – Harry May 28 '19 at 16:31
• I think you may be confusing two forms of notation, but it's hard to tell: in any event, what you wrote does not seem to characterize independence at all. Thus, a good place to begin on this problem would be with a review of the definition. – whuber May 28 '19 at 16:52

Note that $$f$$ is just the $$\text{sign}$$ function (except at 0, but it makes no difference whether you define $$f(0)$$ as $$-1$$, $$0$$ or $$1$$ for this).
That they're individually (i.e. marginally) standard normal is obvious - you're taking a standard normal r.v., and replacing the original sign of it by a random sign ($$\pm 1$$), independent of the original variable.
Now consider the sign of $$VW$$. This should give you some clues about how to proceed.