$X,Y,Z$ 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.

  • $\begingroup$ What ways do you know of characterizing independence? Which have you tried to apply? $\endgroup$ – whuber May 28 '19 at 16:13
  • $\begingroup$ It is $U=|X| f(Y), V=|Y| f(X), W=|Z| f(X)$. $\endgroup$ – Harry May 28 '19 at 16:31
  • $\begingroup$ 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. $\endgroup$ – 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.


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